Talk:Function (mathematics)/Archive 11

Function (set theory)
If we remove a few lines in bottom, then this page is perfect for Function (set theory) instead of Function (Mathematics). The bottom section where people hardly notice is (imho) an accuracy dispute, since when do we agree that function has to has something to do with set and set theory? And the informal concept of function becomes history? --14.198.220.253 (talk) 10:56, 9 November 2013 (UTC)
 * I think the situation is practically completely opposite to what you say. It is mainly in the foundations like set theory where people use functions without a given domain and codomain. Dmcq (talk) 12:23, 9 November 2013 (UTC)
 * I say the situation is independent of what you think. It is mainly in the foundations like set theory where people use functions with a given set of domain and codomain. --14.198.220.253 (talk) 12:52, 9 November 2013 (UTC)
 * That is simply asserting your opinion again. Either provide evidence that shows the article should change or wait for more opinions on this talk page to back your opinion. Dmcq (talk) 14:22, 9 November 2013 (UTC)
 * The same goes for you, that is why i see it as an uncited opinion and turn the "joke" back on you, here is your comment, you can verify it. --14.198.220.253 (talk) 14:25, 9 November 2013 (UTC)
 * You're the one wanting to change things. Dmcq (talk) 14:29, 9 November 2013 (UTC)
 * I don't know how one can explain the previous edit then, everything is an edit on Wikipedia. --14.198.220.253 (talk) 14:35, 9 November 2013 (UTC)
 * Many edits are not objected to and consensus can be gained at the talk page for some which are objected to. Dmcq (talk) 14:42, 9 November 2013 (UTC)

Discuss reverted change casting doubt
Please do not just try forcing things into an article when another editor objects and gives reasons. Please read WP:BRD. Sources say that is how it is said. Provide a source saying it is said some other way. Dmcq (talk) 14:33, 9 November 2013 (UTC)
 * First, BRD is not a policy. Second, is previous edit a reliable source? If not, then an edit has to has a source as long as it is reverted? Where is bold? You see.
 * Third, you think it is "casting doubt", I don't think it is. Let me explain further,
 * According to your edit,
 * $$\begin{align}

f\colon \mathbb{N} &\to \mathbb{Z} \\ x &\mapsto 4-x. \end{align}$$
 * The first part can be read as:
 * "f is a function from $$\mathbb{N}$$ (the set of natural numbers) to $$\mathbb{Z}$$ (the set of integers)" or
 * "f is an $$\mathbb{Z}$$-valued function of an $$\mathbb{N}$$-valued variable".
 * Your edit changes from "The first part can be read as:" to "The first part is read:".
 * When you say "is read", it is not clear that whether it has to be read that way or not. For example, "a function f from N to Z.", "a function f maps from domain N to codomain Z", "f is a function has domain N and codomain Z.", "f colon N arrow Z is a function f from N to Z", Are they all wrong?
 * Is there a transformation rule to those two forms? What makes them so special? Why should it be in those two forms and keep all the words and structure in check?
 * That's why I think "is read" is in fact "casting doubt", or in good faith, unclear.
 * I think you can say $$f\colon \mathbb{N} \to \mathbb{Z}$$ means "f is a function from $$\mathbb{N}$$ to $$\mathbb{Z}$$", you can at best suggest people to read it in certain way, you can say it in different way in English even though it makes use of the word "from" "to" "function"..etc. but you don't say "it is read" because it sounds like it has to be read in the following way. --14.198.220.253 (talk) 15:19, 9 November 2013 (UTC)
 * You changed the 'is read as' to 'can be read as'. The 'can be' means there are other forms also. Have you found other forms? Dmcq (talk) 15:33, 9 November 2013 (UTC)
 * Have you read my post? --14.198.220.253 (talk) 15:35, 9 November 2013 (UTC)
 * Yes. Have you found a citation for a new version? Dmcq (talk) 15:48, 9 November 2013 (UTC)
 * Have you read the article? Have you found a citation for the old version? Yes?--14.198.220.253 (talk) 15:51, 9 November 2013 (UTC)
 * If you are doubting those forms then put in citation needed for places where you require a citation and fill in the reason. Dmcq (talk) 15:59, 9 November 2013 (UTC)
 * Thanks, it is a useful advice, but I am not doubting those forms.
 * Thought you said something about consensus above. You didn't respond to my examples yet, "a function f from N to Z.", "a function f maps from domain N to codomain Z", "f is a function has domain N and codomain Z.", "f colon N arrow Z is a function f from N to Z", Are they all wrong? --14.198.220.253 (talk) 16:14, 9 November 2013 (UTC)
 * They're descriptions of a function f okay, but are they readings of what's written or descriptions of f which just mean the same? 1+1 is the same as 2 but it is read differently. Saying f is a function from domain N to codomain Z for instance is saying N is a domain whereas it is a function from N to Z which has domain N and codomain Z. The alternative definitions section goes into the difference between a function and a function from N to Z and the various ideas of what a function might be. It is certainly okay mostly to just say function here but reading it that way sounds problematic to me. At best it sounds like loose language to me and there's enough loose language already around functions. So yes I rally would like to see someone saying it is read one of those ways rather than you making them up. Dmcq (talk) 16:39, 9 November 2013 (UTC)
 * I didn't say "f is a function from domain N to codomain Z", but even if your interpretation is right, we can just take that one out. So, why can't I read it as "a function f from N to Z"? Is it wrong?
 * It is certainly okay mostly to just say function here but reading it that way sounds problematic to me. At best it sounds like loose language to me and there's enough loose language already around functions.
 * You see, you just said there are "good read" and "bad read", so there is no "must read" or "is read". If you have some kind of transformation rule, for example, to read $$f\colon \mathbb{N} \to \mathbb{Z}$$ literally, you map each symbol to an English word, "f colon N arrow Z". If you can do that for its semantics too, then we can discuss.
 * "is read" is inaccurate for now, we should change it to "can be read", "it is recommended that" "it means"..etc. --14.198.220.253 (talk) 17:33, 9 November 2013 (UTC)
 * Please provide a citation saying is is read otherwise. We are supposed to summarize what is out there, not have people arguing about their own ideas and then stick them in. Dmcq (talk) 19:01, 9 November 2013 (UTC)
 * I have now had a look around at some places where they say how it is read and there are some variations that justify not being dogmatic about it so 'can' can be put in so I'll self revert. Dmcq (talk) 20:00, 9 November 2013 (UTC)

Letting logic prevail over dogmatism
Wikipedia is meant to serve its readers. Insisting on personal opinions is acceptable for mathematics only if they are based on logic, because logic serves clarity in the interest of the readers. Insisting on personal opinions that violate logic is dogmatism, leading to endless discussions and creating only confusion.

This especially holds for the function concept, and it is saddening to see how an essentialy very simple concept is made bewildering to readers by violating logic.

(A) Let's consider the logical consequences of the definition in the main article, here quoted literally:


 * "A function f from X to Y is a subset of the cartesian product X &times; Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset."

Like any definition, this definition consists of two parts:   The definiendum (the concept being defined): "A function f from X to Y ".  The definiens (a defining statement): "a subset of the cartesian product X &times; Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset."  Example:
 * The set {(0, a), (1, b)} is a subset of {0, 1} &times; {a, b} satisfying the condition that every element of {0, 1} is the first component of one and only one ordered pair in {(0, a), (1, b)}. Hence it satisfies the definiens and therefore, by definition, is a function from {0, 1} to {a, b}.
 * The same {(0, a), (1, b)} is a subset of {0, 1} &times; {a, b, c} satisfying the condition that every element of {0, 1} is the first component of one and only one ordered pair in {(0, a), (1, b)}. Hence it satisfies the definiens and therefore, by definition, is a function from {0, 1} to {a, b, c}.

More generally, since the definiens contains the clause " every element of X ", from a given function f we can derive the set X stated in the definiendum. Hence this set is an attribute of f, and therefore can properly be called the domain of f.

The definition does not allow such a conclusion for Y. Indeed, let f be a function from X to Y, and let Z be a set that includes Y but is not equal to Y. Clearly f is a subset of the cartesian product X &times; Z such that every element of X is the first component of one and only one ordered pair in f. Therefore, by definition, f is (also) a function from X to Z. Hence, given the definition, Y is not an atribute of f, and therefore cannot properly be called the codomain of f, and doing so is inconsistent. The most one can do is letting Y be part of the context of discourse and call it the codomain (or the codomain considered), but without the addition "of f".

Recall that the set of first components of the pairs constituting a function f was called the domain of f. The set of second components is also uniquely determined by f and and therefore can properly be called the range of f or the image of f (the choice of terminology being a matter of preference, not logic).
 * For instance, the range (or image) of the function {(0, a), (1, b)} is {a, b}.

The notation $$f\colon X \rightarrow Y$$ is defined by the ISO 80000-2 standard of 2009 on page 15 as specifying that f has domain X and range included in Y. Inconsistency with the definition is avoided by not mentioning codomains at all.

(B) The observations in (A) show that the Wikipedia article as it stands contains many inconsistencies. Let's consider the options to make it correct, apart from the simplest way just mentioned (avoiding codomains)

The only consistent way to make codomains a function attribute is changing the definition, redefining a function as a pair f := (F, Y) or a triple f := (F, X, Y), where F is a set of pairs as defined in the original definition. Bourbaki briefly considered this option on page 76 in Theorie des Ensembles. However, on the very next page (p. 77), Borbaki announces his intention to use the simplest possible definition, also found in the subsequent literature (Apostol, Suppes, Tarski, Mendelson, ...): a function is a set of ordered pairs no two of which have the same first component. Following this definition, domain and range are defined as the sets of first and second components respectively, and $$f\colon X \rightarrow Y$$ as in the ISO 80000-2 standard.

Furthermore, codomains have disadvantages only: they unnecessarily complicate the definition and are very restrictive. For instance, all texts attaching a codomain to a function define the composition $$g \circ f$$ of $$f \colon X \rightarrow Y$$ and $$g \colon U \rightarrow V$$ only for the special case where U = Y. More common definitions, in particular nearly all analysis texts, do not impose any restriction but, instead, define the domain of $$g \circ f$$  as the set of all x in X such that f(x) is in U.

Conclusions: (A) The inconsistencies in the current article have been demonstrated. (B) Among the options to correct them, one is clearly preferable over the other.

Since I will be away during Jan. 12-20, I will not follow the discussions during that period.Boute (talk) 13:23, 11 January 2014 (UTC)


 * You have already made the same argument above, and it has already been addressed. Since there is nothing new in this section except another long repetition of your argument, I don't see any reason for a detailed reply. Please re-read the rest of this talk page. &mdash; Carl (CBM · talk) 14:37, 11 January 2014 (UTC)
 * Untrue: only the conclusion is the same; the argument is formulated in a quite different way, to find out the best way to explain the obvious conclusions to those who are unwilling to accept them. What has remained the same are the reactions: the unwillingness to address the logic: never were any counterarguments given to the main point, illustrated by the (new) example: by definition, {(0, a), (1, b)} is a function from {0,1} to {a, b}, but also from {0,1} to {a,b,c}.Boute (talk) 17:49, 11 January 2014 (UTC)
 * I think the section title says it all as far as Wikipedia is concerned. Yes talking about various ideas of how things should be done is interesting but this is a mathematics article so should follow mathematics sources, and as part of Wikipedia it is subject to WP:OR and WP:V. We need your ideas clearly outlined and obviously so in reliable sources with weight and then it should be documented with its due weight compared to the other stuff around. You have also come at the wrong time as far as I'm concerned as there's a person over at the exponentiation article giving reasons why we shouldn't describe the current state of things as far as zero to the power of zero is concerned, they say it must be defined as 1 in the article. Yep we describe two different ideas which are to an extent contradictory but that's what out there. I don't see this as a much different case with its alternatives section. Dmcq (talk) 16:12, 11 January 2014 (UTC)
 * Also an incorrect appraisal of the situation: the number of references I have provided in these pages for books where functions were presented fully consistently are a multiple of what anyone else has provided. So the bureaucratic pretexts quoting WP:OR and WP:V do not even apply. The argument "that's what out there" indeed justifies presenting different views, but for mathematics articles this is responsible only for views that are internally consistent, and the separation is made clear.  The current article indiscriminately mixes (at least) two mutually inconsistent views without a clean separation, e.g., as I outlined in (B) above. Again (this is the only repetitive part): finally address the logic in the arguments.Boute (talk) 17:49, 11 January 2014 (UTC)
 * Yes but doesn't cut it for Wikipedia. The answer is simply yes we should present the different versions with due weight. We should try for intelligibility too but that is not a Wikipedia policy. I think you may have the mistaken idea that a lot of good logic helps your argument. It doesn't. What is best is reliable sources that clearly show things without needing a lot of extra arguments to explain why what they say shows something. Dmcq (talk) 18:22, 11 January 2014 (UTC)
 * I'm sorry to hear that intelligibility is not Wikipedia policy. One wonders what is left of Wikipedia in that case. Anyway, I agree about needing good sources: they exist in abundance for any topic in mathematics, and I listed many of them on these pages.  However, the simplest and clearest account of function, found in Bourbaki, Apostol, Suppes, Mendelson etc. was systematically vetoed by the editors of this article.  With the risk of being accused of repetition: this definition states: "A function f is a set of ordered pairs, no two of which have the same first component."  Following this definition, domain and range are defined as the sets of first and second components respectively, and $$f\colon X \rightarrow Y$$ as in the ISO 80000-2 standard, namely, specifying that the domain of f is X and the range of f is included in Y.  This definition is fully equivalent to the current Wikipedia definition but, being simpler, creates less confusion in peoples' minds.  I'm off now until the 20th, and then I'll make an edit to that effect.  A note on alternative views with sources will be included, but restricted to self-consistent ones.  Here only logic can tell which is which: we won't find reliable sources stating that 17350983 + 1395618 = 23283568 is wrong. Boute (talk) 21:51, 11 January 2014 (UTC)
 * As explained in the article and in more detail in the article Range (mathematics) the term range is avoided in the article because different authors use the term for different things so it causes confusion. Dmcq (talk) 22:30, 11 January 2014 (UTC)
 * Terminology is the least of the problems and trivial to fix in principle. Yet, I have seen all the alternative terms in the literature, and all them have been used somewhere else for other meanings. It is not hard to accept that every author clearly announces his choice among the many widely used terms, and uses it consistently. Boute (talk) 06:55, 12 January 2014 (UTC)

(Margin reset) The main inconsistency that damages the article --- Irrespective of editors' personal "preferences", is it still unclear to anyone that, given the current Wikipedia definition (undisputed, because free of inconsistencies)


 * "A function f from X to Y is a subset of the cartesian product X &times; Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset."

precludes later introduction of


 * A function f with domain X and codomain Y is commonly denoted by $$f\colon X \rightarrow Y$$

on penalty of inconsistency? And that the ISO 80000-2 standard (or equivalent definitions for $$f\colon X \rightarrow Y$$ from the literature) offers a consistent alternative?

If not, we should all concentrate on finding out why some people fail to recognize this. I dare hope that Dmcq's remark about "reliable sources" does not mean that he wants to see such a source specifically point out this inconsistency in the current Wikipedia article! That is where only logic can settle the issue. Boute (talk) 06:55, 12 January 2014 (UTC)
 * What you are basically saying is that a function from X to Y is a function from X to Z if Y is a subset of Z because it is a set. Yes it is a set but it is a set with an associated type as far as many people are concerned. A number may be defined using a set so we see definitions saying 0 is {}and 1 is –, others use sets for rationals and reals. However the sets so used have types and if we just wrote  – +{} it isn't at all clear a priori what the answer should be without saying what is the type of + or of the sets.
 * The main part of the article takes no sides on this point. That is for the alternatives section because different people disagree on that point. If someone asks for a function from X to Y the X and Y are counted as part of what they want. They don't expect to have to check the values of the function they are given to see if it satisfies the requirements. If a function from X to Y is the same as a function from X to Z then how can one be sure about the function except by checking the actual values rather than the definition? This is the reason some people consider a function from X to Y as having the X and Y as part of the thing being defined. The f so defined may look very similar to another f from X to Z but it is not the same as the function from X to Z as far as they are concerned. Dmcq (talk) 13:20, 12 January 2014 (UTC)
 * It is clear what you are saying but, as the definiendum/definiens analysis shows, the function concept as defined has only X associated with it, not Y. So the only compatible interpretation of the type specification $$f\colon X \rightarrow Y$$ is the one in the ISO 80000-2 standard (or any equivalent formulation).  Hence, by the chosen definition, the article has already taken a clear (and very wise) position!  If you want Y to be part of the function, you have to change the definition into f = (F, Y) or f = (F, X, Y) as in Bourbaki, who reverted on the next page to the simpler definition (wise, for reasons given).
 * From the practical viewpoint, you are also right that knowing Y is useful, but for that it is not necessary to make Y a function attribute. It suffices introducing the function together with the type specification $$f\colon X \rightarrow Y$$. The type $$X \rightarrow Y$$ is just the type of all functions with domain X and range included in Y (see e.g. Meyer's book on programming language theory). Boute (talk) 16:44, 12 January 2014 (UTC)
 * As a practical matter what is covered by the main part of the article covers the majority of cases, but I do think the alternatives section could be expanded to explain the different points of view better. Dmcq (talk) 18:10, 12 January 2014 (UTC)

Property that f^n(x) = id(x)
Is there a name for a function with the property that


 * 1) f^2 = f(f(x) = id(x)
 * 2) f^n = id(x) for some n

where id(x) is the identity function? 130.236.61.243 (talk) 14:19, 6 February 2014 (UTC)
 * (1) is an involution. —David Eppstein (talk) 16:20, 6 February 2014 (UTC)

What is the point of the well-defined link?
I really cannot see what the purpose of linking ' each input gives only one output' to well-defined. This is a functional relation not something about logic. The English meaning is perfectly straightforward and okay. It isn't like 'relation' which means something mathematical. Plus the well-defined article strikes me as a not so well defined article that meanders around the place and will only serve to confuse. Dmcq (talk) 08:08, 30 April 2015 (UTC)


 * Not something about logic??? Everything in mathematics is about logic.  And the idea that 'relation' "means something mathematical" and "function" doesn't seems very strange.Rick Norwood (talk) 12:09, 30 April 2015 (UTC)
 * I didn't say 'function' was not a mathematical term here. I said 'each input gives only one output' was straightforward English and linking it to well-defined would only serve to make people think it wasn't. Dmcq (talk) 12:50, 30 April 2015 (UTC)
 * I will remove that link again. Read then tell me exactly why sticking in well-defined helps. Dmcq (talk) 13:12, 30 April 2015 (UTC)
 * I think what might be okay is to describe the usage of saying a function is well-defined. That could be done in the Definition section or under Specifying a function. Perhaps something like: Checking a function is well-defined means proving that it actually is defined over its whole domain and that each input value gives rise to exactly one one output. For instance $$f\colon \mathbb{Q} \to \mathbb{Z}, p/q \mapsto p+q$$ is not a well defined function as f(1/3)=4 and f(2/6)=8 but 1/3=2/6. There's a citation at the well-defined article that could be used to support this. Dmcq (talk) 13:57, 30 April 2015 (UTC)


 * I agree that putting "well-defined" as a piped link from "each input gives only one output" is a misleading abuse of language. Relations that are not functions are no less "well-defined" than functions.  Let's just say what we mean, and not rely on a confusing article to do that for us.   Sławomir Biały  (talk) 14:07, 30 April 2015 (UTC)

Exposition of terms
Excluding italics, we presently have this:

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function).

Why is the unambiguous (and seemingly modern) term "image" relegated to a parenthetical? I'm thinking of something more like this:


 * In modern mathematics, a function is defined by its set of inputs, called the domain.
 * A set containing the set of outputs, and possibly additional elements, as members, called its codomain.
 * The set of all input-output pairs is called its graph.
 * The specific set of outputs is called the image of the function.
 * Historically, the image of a function is often termed the range of the function; confusingly, in some extant usage, range is instead synonymous with codomain.

Maybe I've overdone the historical angle, but would this not be a more direct exposition? &mdash; MaxEnt 15:57, 30 January 2017 (UTC)


 * I've no objection, go for it. Dmcq (talk) 16:57, 30 January 2017 (UTC)

Removing a paragraph of the lead
I have removed the following paragraph of the lead:
 * The input and output of a function can be expressed as an ordered pair, often denoted (x, y), such that the first element is the input (or a tuple of inputs, if the function takes more than one input), and the second is the output. In the example above, f(x) = x2, we have the ordered pair (&minus;3, 9). If both input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function.

This edit has been reverted by with the edit summary "restore rather fundamental view of function".

I agree that the view of a function as a set of pairs (input, output) satisfying some conditions is rather fundamental. Without the conditions, this view does not makes any difference between a function and a relation, and detailing these conditions here is too technical for the lead. Moreover, this paragraph is confusing as suggesting wrongly that (x, y) is a standard notation for a function. Also, the first sentence is a tautology, as, rewritten in a correct mathematical language, it means nothing else than The pair formed by an input of a function and the corresponding output is an ordered pair. For these reasons, this paragraph has not its place in the lead, and must be completely rewritten before being placed elsewhere. Therefore, I'll remove it again. 17:25, 17 February 2017 (UTC)

Merge proposal
I'm proposing that Empty function be merged into this article (Function (mathematics)). The text there is very short and I think the main article could benefit from a short section describing the empty function. Comments are welcome, and if people agree (or don't object), I'll do the merge in the future. --Deacon Vorbis (talk) 15:08, 22 June 2017 (UTC)


 * ✅ --Deacon Vorbis (talk) 16:16, 25 July 2017 (UTC)

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Definition of a function
The text states that "every element of X is the first component of one and only one ordered pair in the subset.[4] In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f."

If I define a function f: R->R and f(x) = 1/x, this function is not defined at X=0. The above definition implies my f is not actually a function or that the range of f is not actually R, but rather R with 0 removed. Whatever is needed to clarify this situation should be added to the text.

Thanks. Cwm9 (talk) 10:36, 19 March 2018 (UTC)


 * The set X is the domain of the function. The domain of f(x)=1/x is the set of non-zero numbers.  As for the range of a function, many functions have a range smaller than R, which is usually called the codomain.  For example y = x^2 has range the non-negative real numbers. Rick Norwood (talk) 21:24, 19 March 2018 (UTC)
 * But writing a domain as all of R even when a function is undefined in places can be a common error, so it might be reasonable to add something explaining this in a bit more detail. I took the original comment to be looking for something like this.  --–Deacon Vorbis (carbon &bull; videos) 21:56, 19 March 2018 (UTC)
 * I agree with . However more work is needed than adding a simple sentence. In fact, the whole article is written by emphasizing on combinatorial functions. A witness of this is that calculus is not linked to in this article although, for most readers the functions are the functions that are considered in calculus. Another witness is the errors in the second paragraph where it is said that a picture or a list of sample points can define a function. Also it is never said that using a picture of the graph for visualizing a function is, in general, possible only for function with a finite domain and for functions of a real (or complex) variable with specific properties (continuity and differentiability). Thus, a major rewrite of this article would be useful. D.Lazard (talk) 09:05, 20 March 2018 (UTC)

This article is rude to non mathematicians mentioning category theory
The first part is clear for a high school reader, but some parts mention the category of sets, that is too rude to talk about categories and morphisms. I modified the section about the empty function, just mentioning that category theory is a generalized theory of functions to advice the reader that it is other topic. But I was tempted to erase it. Maybe a better option is to separate everything related with category theory to an special section, and review the resulting article with the category theory entry which should include a section on the category of sets aka category of functions. For that the part of function composition should prepare the reader with some properties that are needed in category theory, giving examples to provide an intuitive understanding of how composition can be used to investigate some properties of functions. For example when $$f\circ g=g\circ f=id$$, and others that I don't remember now. — Preceding unsigned comment added by 2806:107E:C:325F:218:DEFF:FE2B:1215 (talk) 08:12, 6 April 2018 (UTC)
 * Good point. As the empty function is a technical concept that is not used elsewhere in the article, I have moved this subsection toward the bottom, making easier for a reader to skip it. D.Lazard (talk) 08:46, 6 April 2018 (UTC)

need to organize it better and give clearer examples in an homogeneous notation
See the example:f: ℤ → ℤ → ℤ = x ⟼ y ⟼ x·y in the currying section. Even for me, knowing what currying is, is obscure. That section talked about the association of the arrows, I expected something like:
 * f: ℤ → ℤ → ℤ = f: ℤ → (ℤ → ℤ)

maybe as an example
 * curry((x,y) ⟼ x·y)=x ⟼ (y ⟼ x·y)

and I still find the later obscure. Isn't more clear if it is explained
 * f(x,y)=x·y curried it is some f"(x)=y ⟼x·y

Although the last mixes different notations.

Other rude mention is the mention to bra-ket notation from quantum theory.

Another alternative could be to split this article in one general (for non-mathematicians) and other advanced (for more mathematic oriented readers) with better explanation of category theory and lambda calculus, as it would be a part of the category theory article, but more detailed. — Preceding unsigned comment added by 2806:107E:C:325F:218:DEFF:FE2B:1215 (talk) 08:42, 6 April 2018 (UTC)
 * I agree with your remarks about what goes wrong in this article, but not about your suggestion of splitting the article.
 * In fact, the article is well developed and (mostly) correctly written for the set theoretic point of view on functions (some exceptions are the bra-ket notation and the "duality between a function and its argument", which have nothing to do where they appear). But, as soon asother point of views are considered, that is from section "Types of functions" on, this article becomes a mess. IMO, the most important missed point is that, for many readers, "function" means essentially continuous function, differentiable function or smooth function. Because of the importance of these concepts in all science, these kinds of functions must be considered sooner in the article, with a clear guide for redirecting the reader to the article to which it may be interested in.
 * For example, it should be said, when discussing about the domain, that in many cases, typically for a function of a real variable, the actual domain is often not given when defining a function and expliciting the domain may need a difficult computation (computing the zeros of a denominator, or the values for which the argument of a logarithm is positive, for example). Although this is fundamental when starting studying calculus, this appears only near the end of the article.
 * Another example is the case of multivariate functions, which are considered in three different sections (Multivariate functions, Currying, Functions with multiple inputs and outputs), without any mention that these sections have the same subject, viewed from slightly different points of view.
 * Good luck for improving the article. D.Lazard (talk) 09:42, 6 April 2018 (UTC)

Rewriting this article
The three preceding threads show that a major rewriting of the article was needed. I have started this, and this is done for more than half the article. In this rewrite, I try to follow the following principles There are probably some other points, but, just now, I am too tired for remember them. Thus I'll add them only if somebody disagree with some part of my edits, which are not sufficiently explained by my edit summaries and the above principles. D.Lazard (talk) 22:19, 13 April 2018 (UTC)
 * Functions are complex objects. It is thus impossible to cover the subject at a totally elementary level. However it must be understandable for everybody who encounter functions in any area of mathematics and applications to other sciences.
 * As Wikipedia is not a textbook, it is not useful to give too much details on trivial examples: if such details could be useful to someone, he does not knows enough mathematics for really understand the article.
 * Most readers do not know of foundations of mathematics, category theory, advanced combinatorics, computability theory, etc. So references to these subjects, including direct product of sets, have been or will be either removed or kept to the minimum. A reader interested to these subjects will probably never start studying them from this article.
 * A similar importance should be given to functions of calculus and functions of set theory and combinatorics. In fact, they are the same functions, but the most important properties are not exactly the same.


 * I'm glad to see you working on this article, but I have to say I can't make out the details on the image you just edited, and I find the stuff under "function" at the top of the page equally opaque. Rick Norwood (talk) 19:06, 14 April 2018 (UTC)
 * I agree with both remarks. I do not like the image, which makes things more complicate than they are, but I have not searched for a better image. The navbox clearly needs to be deeply edited, but this is another task. D.Lazard (talk) 08:37, 15 April 2018 (UTC)


 * Me too. Plus could the functions template used at the top be updated to have something that looks more like a function in it thanks. Also I'm a bit sad that currying was removed, it is very strongly related to functions and tells people a bit more about what can be done with functions. Dmcq (talk) 10:19, 15 April 2018 (UTC)

D.Lazard: Note that begin{align} and end{align} to not have a space before the argument. Rick Norwood (talk) 11:04, 18 April 2018 (UTC)

Recent edits
I have recently edited the whole article for a better covering of the subject. In fact, previous version gave a undue weight to the abstract set-theoretical aspect, with respect to the calculus aspect.

has edited my version, introducing useful improvements as well as some errors and some explanations that are more confusing than clarifying. I have tried to keep the improvements and to fix the issues. This took me several hours of work.

So, please, if you edit the article, try to separate, in different edits, minor rephrasing and structure changes, in order to make easier to distinguish edits that may require discussion. Also, when adding a new section or a new paragraph, please do not edit meanwhile the existing part of the article. D.Lazard (talk) 16:38, 27 April 2018 (UTC)


 * , thank you, I greatly enjoyed that you found something useful and, even more, being an improvement in my edits, and beyond this that you took the effort to conserve this. I plead guilty and apologize for my sloppiness regarding the domain of the Gamma-function, but I beg to agree on disagreement for e.g., on my wordings on different "interpretations", necessary for dealing with "numbers" or "shapes" & "colors", respectively, or on which is a better structure for grouping concepts, or what is a "notational convention". I have to confess that I was not fully aware of all the threads in this TP, and I do not expect the possibility of a unified individual perception of confusion or clarity. It's just that I sometimes feel bold.


 * I am concerned that I caused you several hours of work, I did not intend to keep you busy this way. I will try to make your reversions of what you consider to be my flaws easier to handle, but I do not understand your last sentence. Most respectfully for your work, Purgy (talk) 13:27, 29 April 2018 (UTC)
 * I agree that my last sentence was unclear. My demand was simply to edit, as far as possible, only one section at a time. That is, do not do, in the same edit, copy-editing of existing content of a section and expanding this section, nor restructuring and simultaneously editing restructured section(s), nor editing simultaneously several sections. This would allow, if some editor disagree, discussing and/or reverting only the edit for which there is no consensus. D.Lazard (talk) 15:36, 29 April 2018 (UTC)

Alsosaid's edit
Good work on functional notation, Alsosaid and Purgy. Rick Norwood (talk) 10:58, 19 June 2018 (UTC)


 * Thanks! I think it was a good idea to have a section specifically addressing notation. I think there's a question of whether specialized notations should be mentioned in this section (e.g., lambda calculus notation, or bra-ket-like notations like [x,f] that give equal footing to function and argument).  Obviously, the description should be limited to a paragraph or two.  Alternatively, specialized notations could be placed in a later section.  Let me know what you think. Alsosaid1987 (talk) 16:21, 19 June 2018 (UTC)


 * I am not sure about your intentions, but as I see it now, I would object to both lambda calculus notation, as to bra-ket-like notations. As it stands, the article is (luckily?-yes!) not very explicit about binding variables, so I would experience lambda calculus as an overkill. I think it belongs to more formal, more logically oriented articles. I am unsure about the "-like" and the "equal footing". There are now efforts to discriminate (a cognitive merit!) the meaning of function and argument, so I miss the target of equalizing. The bra-ket in itself takes its immense importance -imho!- from incorporating operators and their adjoints within the notation for a scalar product in a specific function space, so -again imho!- is too specialized to be included within a chapter Notation on general functions. As a "See also", perhaps? Purgy (talk) 08:17, 20 June 2018 (UTC)
 * Addendum: Perhaps the scenario often expressed in the \case-environment should get some attention, to get rid of the naive assumption that functions are always easily accessible via an algebraic expression, maximally involving some standard function. Purgy (talk) 13:28, 20 June 2018 (UTC)
 * IMO, lambda calculus notation would be better placed in section "In computer science" (presently reduced to a main template). Similarly, bra-ket-like notations should appear in section "Function space". In fact, these notations are useful only when a function space has a structure that is similar with the domain of the functions (typically, duality in linear algebra). In fact, these notations are useful only for people who are already interested in these special classes of functions. D.Lazard (talk) 15:07, 20 June 2018 (UTC)

Purgy's edit
Thank you for your help. Rick Norwood (talk) 11:33, 4 July 2018 (UTC)

Etymology
By mistake, I have removed the section "Etymology" without edit summary. Here is my motivation.

The non-mathematical etymology of the word function does not belong to the mathematical article, but to Wiktionary (by the way, the etymology given there, is almost identical with the one that I have reverted, and is strongly dubious, as asserting implicitly that the French word "fonction" is not the same as the old French word "fonction").

I learnt from History of the function concept that the concept has been named "functio", in Latin, by Leibnitz and Bernoulli. The translation in English from "functio" to "function" is so evident that it does not deserve any mention in the article. D.Lazard (talk) 13:16, 10 July 2018 (UTC)


 * Yeah, I agree. Any etymological information in a math article should generally be about how the (usually preexisting) word(s) came to acquire its mathematical meaning, and not how the preexisting English word itself came into English.  –Deacon Vorbis (carbon &bull; videos) 14:22, 10 July 2018 (UTC)


 * Thanks for the motivation! I was about to revert the whole edit, when Proto-Indo-European caught my eye, and I searched for a better place, and tried to remove as much rubbish as I could recognize. I think I better understand now what etymologies are appropriate. Purgy (talk) 16:16, 10 July 2018 (UTC)

Definition
My edit changing "A function is defined by..." to "...is defined to be..." was reverted on the grounds that the latter is the graph of the function and not the function itself. However, functions are usually defined to be exactly the same as their graph. As is, this article currently has no actual definition for a function. It's okay if we don't want to dwell on the formalities excessively, but there should be some indication that this is how the intuitive notion is actually formalized. –Deacon Vorbis (carbon &bull; videos) 15:09, 16 July 2018 (UTC)
 * It is true that formally (that is syntactically) a function is the same thing as its graph. However, semantically, the function and its graph are two distinct objects (for an example of this, try to define the derivative (and to explain it to a student) by using the abstract graph only (without the functional notation, and without drawing). This complexity of the concept must not be hidden. I have edited the article as an attempt for clarifying this in a non technical way. My edit may certainly be improved, but, in any case, the fact that a function is generally thought as a process must not be hidden. D.Lazard (talk) 18:25, 16 July 2018 (UTC)

Dual pair notation
Dear Purgy, I think some care needs to be taken when showing the symmetry so as to not be somewhat misleading. Your expression x(f) is really equivalent to evaluation of f at x: f\mapsto f(x) for some fixed x. That is a member of the double dual, V**. There's, of course, a canonical injection x\mapsto (f\mapsto f(x)), an isomorphism for FDVS but treating vector x as if it were a member of V** is an abuse of notation (though a convenient one). Alsosaid1987 (talk) 17:31, 17 July 2018 (UTC)


 * I have made edits on the basis of yours to reflect these points. However, this section is considerably longer than my original intention (since this is a special purpose notation).  I purpose that this notation and the function notation from the lambda calculus be mentioned with links but not necessarily described in length (if at all). What say you (and others)? 150.212.127.46 (talk) 19:43, 17 July 2018 (UTC) It's me. Alsosaid1987 (talk) 19:44, 17 July 2018 (UTC)
 * I agree with . I must say that I have rewritten the section in March 2018 by these edits. I removed the bra–ket notation, because I was not able to present it in a way that is useful for a normal reader (a reader interested in functions may be concerned by this notation only if he is also interested in linear algebra. In this case, he will be able to get relevant information in linear algebra articles). This is the reason of my choice of not mentioning this notation. Maybe I was too bold. But, IMO, a simple sentence, with links is sufficient is one want to avoid to be too technical in this section (at the beginning of the article). D.Lazard (talk) 20:51, 17 July 2018 (UTC)
 * Dear Alsosaid1987, I am aware of at least some subtleties regarding the double duals of vector spaces, and these subtleties were among other aspects a reason for my reservations against mentioning "bra-ket-like" notations (see above "Alsosaid's edits") in this article. My edit, which caused this thread, intended to explain the postulated "symmetry", and I tried to take care of the subtleties by using the phrase "Under certain premises ..." I think without this "abuse of notation" there is no obvious "symmetry". refactored 07:15, 19 July 2018 (UTC) Purgy (talk) 11:18, 18 July 2018 (UTC)
 * I agree. After some thought, I felt symmetry is less apt of a description, since it only comes from the "abuse of notation".  The "abuse" is enlightening and is mathematically meaningful because of the natural homomorphism, but in itself, it is not fundamental. Alsosaid1987 (talk) 14:50, 18 July 2018 (UTC)


 * Given the article in its state of now, I have just a slight desire to remove "and symmetry" from the phrase "underlying duality and symmetry" in the chapter "Specialised notation". I am not sufficiently authoritative, whether the "commutative diagrams" are really notations of functions and belong there, or whether they just employ arrows and names of functions, and are better removed. Purgy (talk) 11:18, 18 July 2018 (UTC)


 * I guess you're right about duality and symmetry, since they describe the same phenomenon. I will defend my position on commutative diagrams, since showing commutativity is not an obvious extension of the notation.  Moreover, the use of several types of arrows to show mono, epi, iso, existence (dashed arrow), and uniqueness (!) are also not properly part of arrow notation, at least as described above.  Alsosaid1987 (talk) 14:30, 18 July 2018 (UTC)
 * Maybe an article about prevalent(?), contemporary use of the several available arrows (barred, hooked, double, wavy, harpooned, hatted, somehow rotated, dashed, ...) is missing in WP. I think nevertheless, that the networking arrows (morphisms?) are about relations between certain maps, and do not necessarily contribute to the specification of one of the involved maps, which possibly is not a function at all. Afaik, these diagrams rather are used to readably formulate properties (subject to a proof) of the construction established by these maps. Purgy (talk) 07:15, 19 July 2018 (UTC)
 * About barred, hooked, double, wavy, harpooned, hatted, somehow rotated, dashed, ... arrays: None has a meaning that is universally accepted. This means that any author who want use it must define it. Therefore, there is no encyclopedic content that can be added to WP. The only exception is $$\mapsto,$$ which is described here.
 * About commutative diagrams: They mean equality between composition of functions, and this means that it is worth to mention them in this article. However, the arrows that appear in them is not a specific notation, it is the common function notation with arrows. Thus, I would tend to remove them from the notation section, and add a few words in the section about composition. IMO this deserves to be discussed further. D.Lazard (talk) 10:57, 19 July 2018 (UTC)
 * I don't know that the harpoon (double-pointed) arrow for epi and (back-)hooked arrow for mono are non-standard. Not being a mathematician, I don't know the exact details, but I've seen paper where knowledge of the meaning of these symbols is assumed, so some journals evidently don't require them to be defined.  Moreover, they've been exported to other branches of math.  I agree that these things are better left off of this article, but they should (and do) appear on the articles on commutative diagrams and surjective and injective functions. My own field of chemistry has its own peculiar rules regarding arrow usage, so starting a new article like that for math seems to invite an article like that for chemistry as well...  I'm not against it, but to write articles like that well and in an encyclopedic style is not easy. Alsosaid1987 (talk) 00:49, 20 July 2018 (UTC)


 * My cursory remark about a new article on arrows was just a negligible aside, I am but a bit serious about the diagrams not belonging to notations for functions, but being a notation for describing "interactions" of maps, based on the functional composition, and thus belonging to this section, if at all to this article. Purgy (talk) 06:10, 20 July 2018 (UTC)
 * I don't disagree that it is not a type of function notation per se, but it is highly related to the arrow notation. I would be fine with a brief reference to it as an extension of arrow notation to describing relationships between functions. Alsosaid1987 (talk) 17:49, 20 July 2018 (UTC)

Definition... again
This comes from the Vertical line test page; A function can only have one output, y, for each unique input, x.

Why is this so? It’s not explained on that page, and I can’t see it explained in this page [“Function (mathematics)”]. Nor can I see it in History of the function concept (it might be there but I can’t see it).

And what do you call a (non-function) such as x = y² (or y = √x) which has more than one output for x? Why isn’t that a function? MBG02 (talk) 07:38, 28 July 2018 (UTC)


 * Please, do not take this as offensive: In mathematics some things are so, because they are defined to be so. There is no ontological reason, but only that it turns out to be useful to have these things this way (in lucky cases even "not to have them in an other way"). Of course, usefulness is disclosed only in extended use. A “History of the function concept” just describes how it came along to have quite good agreement among mathematics on this (some physicists and engineers may still cling to more lose notations, afaik, teachers are to blame for this).
 * My interpretation of your non-mathematical term "non-functions" leads to the term "multivalued function", which best should be abandoned for being badly confusing, especially for newbs, but it is there, because it too is useful, especially within complex numbers ... The blatant reason why such relations are no functions is, as said above, because they do not satisfy the definition of a function. Real examples for these are the inverse maps of the trigs (arcsin, ...) and all other periodic functions. Generally, these maps can be restricted to principal branches, which satisfy the functions' properties.
 * On pure formal grounds the standard real √x is a function, because it is defined as a function from the non-negative reals to the non-negative reals, the "negative branch" of the square root is cut off in the standard, generally agreed upon, definition. Calling x = y² a non-function is suggested by an unlucky manifestation of extending a prevalent notation (as is y = f(x)) to inappropriate ranges. The y is too much abused in using it for a value (of the function f at x), and also for the function itself (as in y = y(x)). The map with $$x \mapsto x^2$$ describes the same function as $$y \mapsto  y^2$$, but $$x^2 \mapsto  x$$ is no well defined notation (afaik).
 * HTH. Purgy (talk) 09:37, 28 July 2018 (UTC)

One reason for functions to have a single output is that it avoids ambiguity. If a function has more than one output for a given input, then one person may think one of those outputs is intended by f(x) while another person may think a different output is intended. A good example of this ambiguity is the "function" f(x)=arccot(x), the inverse of the cotangent function. Some books, calculators, web pages, and computer programs say f(-1) = arccot(-1) = - pi/4. But others say f(-1) = arccot(-1) = 3 pi/4. Everyone agrees that this ambiguity is a bad thing, but nobody has the authority to fix it. There is an International Bureau of Standards, but no International Bureau of Mathematics. That is why I put the word "function" in quotation marks above. Each definition of arccotangent is a function individually, the two taken together make the expression f(x)=arccot(x) ambiguous, and threfore less useful. As Purgy said, mathematicians are guided in their definitions by what is useful.

Mathematics becomes less useful when schoolteachers (and physicists) make up their own definitions as they go along. Rick Norwood (talk) 11:12, 28 July 2018 (UTC)


 * Actually there is ISO 80000-2 which does have a section on Circular and hyperbolic functions which includes their inverses. But as is implied in its title "Quantities and units -- Part 2: Mathematical signs and symbols to be used in the natural sciences and technology" they haven't got mathematicians on-board ;-) Dmcq (talk) 11:58, 28 July 2018 (UTC)


 * More precisely re arccot: There are countably infinitely many values, differing by $k$$\pi$, that have the property of being an arccot($x$) for some real $x$. Not agreeing on one interval, in which exactly one of these values for each $x$ make up a true function, the (piecewise) smooth "principal branch", often termed Arccot (note the capital "a"), is the (sad???) standard (mostly $(-\pi⁄2,π⁄2]$ vs. $[0,\pi)$). Personally, I try to avoid using "arccot($x$)" and an "=" together (equating sets is rare with inverse trigs); using the function Arccot and stating its codomain is the way to go for me. Purgy (talk) 13:03, 28 July 2018 (UTC)

I like the “just because” answer most (then the “avoids ambiguity”). The terms multivalued function and single valued function didn’t arise in my searches for a definition (/explanation).

Arc Cotangents are bit hard: got anything x-y that’s easier to understand? MBG02 (talk) 18:38, 29 July 2018 (UTC)
 * I'd suggest not thinking of the graph of a function as it seems to be distracting you from what has been said above. Arccotangent is an inverse trigonometric function and should not be hard for anyone to understand. If you need a more elementary example, try defining an inverse function to any non-injective function.--Jasper Deng (talk) 19:02, 29 July 2018 (UTC)

(In-)dependent variables
Sorry, but I do not want to see the above classification revived (in the lead!) without any discussion. I consider it to be alien to a more modern notion of function, and of no serious help in understanding the notion of function. Primarily, I see these categories as an outdated treatment of inverse functions. Imho, it should be about argument and value, nowadays.

I do not care much about the removal of "process" and "relation", whether something is an image "by" or "under" (I perceive the "by" as more stylish), but "common names" for variables are a no-go to me.

I would prefer not to have the "special names" in the lead, and would like to trim the orgy with common, useful, science and whatnot in the end.

As a result, I cannot do better than revert. ;) Purgy (talk) 15:47, 30 August 2018 (UTC)


 * Sorry you feel that way. I gave it my best shot.  Your turn. Rick Norwood (talk) 18:13, 30 August 2018 (UTC)

Functions of
The article currently claims that it's standard terminology to call a modern function, i.e. a map f:A->B "a function of x". As far as I can tell that makes absolutely no sense and it is also not standard practice. On the other hand it is standard practice to call f(x) a function of x (look up any calculus book from Euler to Courant.) So I would kindly ask the users reverting my changes to back up their claim that "f is a function of x" is standard practice with some evidence. (MBachtold) — Preceding unsigned comment added by Mbachtold (talk • contribs) 06:38, 29 August 2018 (UTC)


 * Please, put new sections at the bottom of a TP (perhaps using the appropriate tab "New section" at the top), and
 * please sign your contributions on TPs, either by writing four tildes (~) yourself, or by clicking the third-left button in the top edit tool bar.
 * BTWs: TPs are the appropriate place to discuss edits of the associated article; in no way feel bullied by being asked if an ADMIN may call you RANDY. Having been bullied by an admin has probability (and credibility) zero.


 * f(x) is not a function. It is the value of a function. The function is f. Maybe thinking about the following simple example will help. Suppose that you wanted to find out the value of the function at the argument 3. Then you would write f(3), not f(x)(3). So in expressions of this form, the function is named f, not f(x). If you write f(x), you get an expression that might have some specific value (if we already know what x is) or that might be indeterminate (if we don't), but that's not the same thing as a function. —David Eppstein (talk) 07:07, 29 August 2018 (UTC)
 * @David Eppstein: I never claimed that f(x) is a function, I claimed that f(x) is a function of x, which is something different. You don't need to belabour the point that f and f(x) are different things, I know that. The first one is the function according to the modern definition, the second one is not. But a modern function is not a function "of anything". For instance the function (x \mapsto x^2) is not a function of x since the variabel x is bound in the expression (x \mapsto x^2) (indeed it is alternative notation for the lambda calculus notation (lambda x. x^2)). So x \mapsto x^2 is the same function as y \mapsto y^2 or t \mapsto t^2. But bound variables don't denote anything, so it wouldn't make sense to say that y \mapsto y^2 is a function of x as the entry currently suggest. But it does make sense to say that x^2 is a function of x. This is why I want to change it to "f(x) is a function of x". (Mbachtold) — Preceding unsigned comment added by Mbachtold (talk • contribs) 07:38, 29 August 2018 (UTC)


 * I sense the heavy pressure towards "standard lingo", but in principle I support your POV. I cannot perceive that upholding the the current formulation "helps understanding" more, but OTOH, requesting a RS for "one often says that f is a function of x" is ridiculous, since IT IS often said, and WP does not state any truths, but just reports what is said in sources that are considered reliable by ... (consensus?). Purgy (talk) 07:49, 29 August 2018 (UTC)


 * Well, if it's ridiculous to ask for evidence that people "often" say "f is a function of x" (which I honestly don't think anyone in their right mind should say), it seems even more ridiculous to me to ban the phrase "f(x) is a function of x", since that is used even more often. Moreover, the phrase f(x) is a function of x is not meaningless, as opposed to the claim that f is a function of x. Mbachtold (talk) 08:08, 29 August 2018 (UTC)


 * Since you reverted my last change without explanation, could you at least provide one citation of a respected mathematician who explicitly calls a modern function $f$ a function of $x$? The page currently cites MacLane Birkhoff but I can't see them doing that. Otherwise please explain why my change was reverted? In case someone here actually needs evidence that the expression “f(x) is a function of x” is used very often and has been correct for more than 300 years of mathematical history, please consult the following discussions and references therein: Who first considered the f in f(x) as an object in itself and who decided to call it a function and Formalizations of the idea that something is a function of something else.
 * The given citation doesn’t say what the article claims. Give me a page number. Moreover I’m not promoting my point of view, I’m promoting the point ov view of Euler, Legendre, Cauchy, Gauss, Riemmann, Gauss, Hilbert etc. as you would see if you read the links above, instead of arrogantly ignoring my points and promoting your own confused point of view. Do people always behave like in a Kindergarden here? Mbachtold (talk) 18:13, 29 August 2018 (UTC)


 * Well, firstly, I shouldn't have written ridiculous, apologies. May I never the less ask you to check the possibility of reading "f is a function of x" as a declarative statement, establishing x as the name for a variable, which is vastly ("as standard", generically, ...) employed to denote a function f with. The f is similarly a standard or generic name for functions as the x is for variables. As usual in informal attempts of explanations, the whole context of the statement and especially the scope of the bounds of the names are not really explicit (neither is domain and codomain of the function).
 * I perceive the tone of your harping on "f(x) is a function of x" as getting increasingly aggressive, while not responding to the claim that f(x) denotes (primarily???) the value of f at x, and not the function f itself. Please, assume my consent to you feeling rightfully annoyed, but this is WP and there are far less welcoming areas then this small TP-discussion, see e.g. even the math project page itself, where freedom of speech is suppressed in a really stinky way.
 * Finally, may I ask for your opinion on my suggestion, in which I tried to resolve the abuse of notation, but which was considered as not helpful in understanding. Purgy (talk) 09:44, 30 August 2018 (UTC)
 * Dear, thanks for your attempts to return to a more welcoming tone. Indeed I'm annoyed by people immediately reverting my changes without addressing the questions and points I raise. Moreover, suggesting that I suffer from Dunning–Kruger effect and that I'm promoting an uninformed point of view, while at the same time showing a lack of understanding for the idea of free and bound variables and for the history of mathematics is not helping this discussion. But I'll behave if others are able to do so too.
 * Turning to your questions: I think the first thing to understand here is that the phrase "a is a function of b" has a very long tradition in mathematics (check the links I gave above) and to a certain extent also a very clear cut meaning (which is not the same as "a is a function" in the modern sense). This established meaning is being distorted by the current statement "f is a function of x" in the article. Your suggestion of reading "f is a function of x" as a statement establishing x as the name for a variable is of no help, I'm afraid. Of course you're allowed to make a statement establishing the name of a variabel, but why would you drag the function f in for that? Can you explain? It's not like the name of the (free) variable x gets somehow attached to the function f by that statement, is it?
 * Next you suggest that I'm not responding to the claim that "f(x) denotes (primarily???) the value of f at x, and not the function f itself". I thought I made it clear that I even agree with that claim, when I responded to David Eppstein. The difference between f and f(x) is the same as the difference between a function and a function of x. (Maybe the linguistic subtleties here are irritating?). In any case, if you believe I suggested somewhere that f(x) is a function in the modern sense, or that it is the same as f, please tell me where and I'll try to clarify.
 * Finally, concerning your suggested edit, although it's a lot better than the current wording, it's still suggesting that a modern function f is a function of something, which as I tried to explain above is wrong/meaningless. I would appreciate if people could directly address my arguments in the response to David Eppstein, or at least tell me what's not clear about them.Mbachtold (talk) 11:51, 30 August 2018 (UTC)

This discussion has been going on for some time, and while the point that f(x) is the output of f at x, not the function, common usage ignores the distinction, especially for named functions, thus people say "the function sin(x)" or "the function ln(x)", just as they say "the number 3" when properly they mean "the number represented by the numeral 3". Still, there is an important point here, and I'm going to make a try at helping readers of Wikipedia understand what we mean when we say "the function sin(x)". Rick Norwood (talk) 14:26, 30 August 2018 (UTC)

Having followed the link by Purgy to some of the discussion around here, I have decided that I don't want to contribute any further. Feel free to ignore anything I have said (I'm sure you already did.) ByeMbachtold (talk) 14:42, 30 August 2018 (UTC)


 * Given the above statement I didn't feel that a response by me was needed. But as you seem to be still involved in editing this page, I've changed my mind about that. There are several things I wish to say, but primarily you must come to understand that Wikipedia is not a debating society, we report on what is in the reliable literature (although sometimes imperfectly) and not on what we think should be there. Your POV, based on the early development of the function concept, seems to be totally devoid of the effect of evolution on mathematical concepts. The early ideas get tweaked and changed over time because they do not fully capture the nuances of the maturing concepts. Advocating a return to earlier views is just ignoring the refining influence of time. Your comments are laced with personal attacks, typically a sign of immature frustration, and that puts so many editors off that you are not likely to make any converts.
 * As to the specific issue on the page, the proper way to deal with a citation that you don't think is correct is to apply a tag with an explanation. Other editors, with access to the source, can either verify or fix the problem. Drive by tagging or removal of citations is viewed as just petulant childish behavior. I do not have access to the work in question, so assuming good faith on the part of the editors who put it there, I reverted your edit. However, I do believe that the statement is not present in the work. It strikes me as a mashup of different traditions in the development of the concept of function. I also think that it is a very common oral expression used by countless numbers of instructors at the calculus level or below. It is a colloquial way of indicating the importance of the domain of a function without getting too abstract for that level of instruction. Such verbal emphasis is very important in teaching, it gives something that will likely stick in the student's mind, but, not being technically correct, will not find its way into written work. Your vocalized inability to parse this expression is just a sophomoric debating ploy (giving you the benefit of the doubt) and holds no credence. --Bill Cherowitzo  (talk) 19:48, 1 September 2018 (UTC)

Quote from Apostol's book.
The quote from Apostol's book (page 35) is

In the section "Further Terminology concerning functions". If S is a subset of D(F) [the domain of the function F], we says that F is defined on S. In this case, the set of F(x) such that x\in S is called the image of S unider F and denoted by F(S). If T is any set which contains F(S), then F is also called a mapping from S to T. This is often denote by writing

Take into account that the addition of the clarification that functions and mappings are concepts that are used as distinct concepts, doesn't contradict or subtract from the content that existed before, in the section. It rather clarifies it, how authors that present the topic more precisely make the distinction, while at the same time, saying that it is common to use the two notions interchangeably. Cactus0192837465 (talk) 13:38, 9 September 2018 (UTC)


 * I do not disagree with any of Apostol's statements, but do so strongly with your phrasings and interpretation of those. Especially, I oppose to your use of the naming concept, which essentially contradicts the concept, which I adhere to, of naming objects and the scope for the meaning of these names. I totally disagree with your claim of X or Y having a general meaning within the universe of this discussion. Furthermore, I am not willing to take "function" and "mapping" as both denoting the same mathematical entity, as used by a majority of math literature.
 * OTOH, seeing your method of reverting and adding other -let's say- dubious stuff without slightest hesitation, when asked to discuss your intentions, and of posting incoherent snippets, possibly of standard textbooks, on the TP, I am not interested in continuing a debate. Purgy (talk) 10:35, 10 September 2018 (UTC)

Doesn't this contradict the definition of "function"?
Under the subhead "Specifying a function", the first sentence is:

"According to the definition of a function, a specific function is, in general, defined by associating to every element of its domain at least one element of its codomain."

The qualifier "at least" makes it sound like a given element of the domain could map to more than one element of the codomain. This would violate the basic meaning of "function," wouldn't it? Or am I missing something?


 * You are right. I've fixed it and made some general improvements to the language of that section.--Bill Cherowitzo (talk) 03:32, 18 October 2018 (UTC)

Another suggestion for an improvement (not a "correction" as such) to the section "Specifying a function": The phrase


 * "When the domain of a function is the set of nonnegative integers or, more generally, when the domain is a well ordered set, a function may be defined by induction or recursion,..."

introduces well-ordered sets as a possibly misleading example of a generalization, since essentially one needs some well founded relation in the domain to support a recursive definition, linear ordering not being at the heart of the matter. I suggest replacing the above phrase with


 * "When the domain of a function is the set of nonnegative integers or, more generally, when any well-founded relation is defined in the domain, a function may be defined by induction or recursion,..."

The page Well-founded relation conveniently has a section describing induction and recursion in that generality. Lapasotka (talk) 08:35, 23 October 2018 (UTC)
 * Formally, you are right, but this seems unnecessary WP:TECHNICAL for this article, as, normally, a reader of this article has never encountered functions with a domain having several minimal elements. D.Lazard (talk) 09:23, 23 October 2018 (UTC)

A confusing line.
>"In general, these points form a curve, which is also called the graph of the function. This is a useful representation of the function, which is commonly used everywhere, for example in newspapers."

I don't really understand this line. What does it mean by "for example in newspapers"? I feel this example raises more questions than it answers, for me at least. — Preceding unsigned comment added by CongealedBox (talk • contribs) 18:15, 4 December 2018 (UTC)
 * Do you have never seen, in your favorite newspaper, the graph of the evolution of a price index? Nevermind, I have made the sentence more explicit. D.Lazard (talk) 18:38, 4 December 2018 (UTC)

Section Representing a function
There are lots of ways to represent a function:


 * Listing of all the pairs.
 * Giving the graph as a plot.
 * Giving a transformation of elements of the domain to elements of the co-domain. For example, a formula in the case of numbers.

and many others. However currently the section only contains Graphs, and one that is wrong, Histograms, which are not representations of a function. For example, one cannot recover a function from a histogram created from it. It would be nice to enrich this section with correct information. Cactus0192837465 (talk) 22:15, 26 December 2018 (UTC)


 * Maybe we should distinguish between "vizualizing a function" and "representing a function"? Plots and histograms can be used for the former, graphs for the latter. In many cases, a function from a finite domain to a finite codomain can be recovered from its histogram (if it has sufficient resolution, which is not the case in the traffic statistics example shown in the article). - Jochen Burghardt (talk) 07:42, 27 December 2018 (UTC)
 * The thing is that histogram is just something else. It neither represents the function, nor visualizes it. It is a transformation, which has its uses like approximating the distribution of data. Allowing transformations in that section opens the door to many others that actually would represent a function even better: Derivatives, integrals, or even symmetries. Let me also quote the second sentence in the article for histogram: 'It differs from a bar graph, in the sense that a bar graph relates two variables, but a histogram relates only one.' The bar graph here would be the function out of which a histogram was constructed. You said that 'in many cases, a function ... can be recovered'. I would try an example before making such a claim. Because what is really the case is that *no function* can be recovered from a histogram, no matter how fine the bins. The reason is precisely the sentence that I quoted above. For example, take the function $$f:\{2\}\to\{1\}$$, which sends the number 2 to the number 1. The histogram of its values would be $$h:\{1\}\to\{1\}$$, since the value 1 is taken once. There is no way to get back any information of who was the pre-image of the value 1, from $$h$$. Cactus0192837465 (talk) 14:05, 27 December 2018 (UTC)

Uff. There is another thing really wrong in that section. There are two images, one of a function of years versus deaths, and then the other supposedly of its histogram. But that second image is not a histogram, but a discretization of the same function. It doesn't output frequencies, but still deaths. I am impressed how such a mistake has lasted for so long in a page like this one. Cactus0192837465 (talk) 14:53, 27 December 2018 (UTC)
 * I understand now the mistake made by the guy who created the supposed histogram. They used set style histogram in gnu plot with the same data file and assumed that that produced a histogram. Cactus0192837465 (talk) 15:05, 27 December 2018 (UTC)
 * Please, do not delete image or content when a slight correction suffices to fix an error. In particular, in the figure you have removed, the change of "histogram" to "bar chart" suffices for fixing the caption.
 * More generally, everything that you qualify as wrong results from a (very common) confusion between bar charts and histogram. I have edited the article for avoiding this confusion.
 * Nevertheless, for being clear, bar charts are useful for visualizing functions with a small finite domain, especially when the elements of the domain are not naturally ordered or correlated. On the other hand, histograms are the only tool allowing a (partial) visualization of a function, whose domain are large finite sets with uncorrelated elements. In fact, a histogram is the graph (generally represented as a bar chart) of the function that maps each element of the codomain to the size of its inverse image.
 * So, please, do not qualify as wrong paragraphs and sections that are simply badly or confusingly written. Instead, either fix what needs to be fixed, or if you are unable to do it, tag the dubious sentence with clarify, citation needed, dubious, or with other inline cleanup tags. D.Lazard (talk) 17:28, 27 December 2018 (UTC)
 * I will delete anything that is wrong as it is better to not have disinformation. The re-addition of that image will be temporary anyway. Once the author (or someone else) corrects the svg to represent a proper histogram, the new caption will be once again wrong.
 * Re:'histograms are the only tool allowing a (partial) visualization of a function'. Histograms don't represent functions. They are a transformation (or if you want a representation) of data. There is no need of domain and image, just a set of data is enough.
 * Re:'whose domain are large finite sets with uncorrelated elements'. This makes no sense. What are correlated elements? This is a rhetorical question. I know what correlation is, but that sentence is so nonsensical with the concepts involved so incorrectly applied that I don't know where to being. Correlation? Of elements? Of a domain of a function? A general function? You were once a mathematician, but this sentence looks written by someone who doesn't have any idea of what any of those concepts mean.
 * Bar chart makes sense as a subsection, as I mentioned the bar chart is the graph of the function. The section could also mention some other representations too. Histogram is just nonsensical. Cactus0192837465 (talk) 18:11, 27 December 2018 (UTC)

Deletion of paragraph
This paragraph

Histogram are often used for representing very irregular functions or functions whose elements of the domain are weakly correlated, as it occurs frequently in statistics. For example, for representing the function that associates his weight to each member of some population, the histogram is the graph, generally represented by a bar chart, of the function that associates to each weight interval the number of people, whose weights belong to this interval.

is a disaster and should be re-written.


 * 1) 'elements of the domain are weakly correlated' should go. This makes no sense. Elements of the domain of a function have no sense in which to be qualified as correlated or weakly correlated, or not.
 * 2) 'his' Functions or histograms are not persons. Modern English doesn't give them a grammatical gender either. This should be 'its'.
 * 3) The last sentence (which is really two or three sentences separated by commas) attempts to explain what is a histogram, but it makes no sense. First it says that the histogram associates weights to members of some population. Then it says that in the graph of the histogram each 'weight interval' (bins it is called) is associated to the 'number of people'? Now there are 'people' in a histogram.

Can someone write a version of this paragraph that at least uses proper terminology, proper grammar, and coherent ideas? I don't want to do it because I think that histograms don't belong in this section. See previous section in the talk. Cactus0192837465 (talk) 21:28, 27 December 2018 (UTC)
 * Some comments:
 * As far as I know, "correlated" means "related together". In any case, this is the etymological meaning. If this sense is no more colloquial, "correlated" may easily be replaced by "related".
 * "his weight" refers to "member of some population". Generally a member of a population is a person, so "his" seems correct.
 * Are you of good faith? I agree that the sentence is too long and should be split, but you present its content in a clearly biased way: the sentence does not "says that the histogram associates" anything. I say that the histogram of a function is something. Here is the split version of the sentence, which you could easily infer if you were of good faith: "For example, let us consider the function $f$ that associates his weight to each member of some population. One can associate to $f$ the function $g$ that maps to each weight interval the number of people, whose weights belong to this interval. A histogram for $f$ is the graph of $g$, generally represented by a bar chart." I do not like this version, as too technical, because of the use of symbols for naming functions, which should be avoided in this elementary part of the article. If you find a better formulation, please, feel free to propose one. However, an advantage of this formulation is that it makes clear that a histogram is a tool for visualizing a function, and thus belongs to this section.
 * D.Lazard (talk) 22:31, 27 December 2018 (UTC)
 * This is a mathematical article, and correlation has a precise meaning in mathematics, specially when talking about histograms, which is also in the area in which correlations is most used. But more importantly, that criterion that you made up, namely functions with many values with elements in the domain that are "weakly related together", has nothing to do with the use of histograms. If you don't know about something, there is no need of adding imaginary explanations. It is better to not have a content than to have something that makes no sense.
 * 'his weight' when referring to a function is also not proper English. It should be 'its'.
 * The sentence is all over the place independently of anyone who reads it. The version in green is not improving it. It has all the same problems. 'weight interval'. That is not called that way. What is a weight? I know (or can guess) what you are trying to say, but it is not called that way. You might be referring to either the bin sizes. But it the explanation is so bad that maybe you are using it to refer to the frequency in which the data falls in a bin. This version in green also emphasizes even more the error that a histogram is constructed out of a function. It isn't. A histogram is constructed out of data. A set of values is enough as input. That is precisely why I am saying that histograms don't belong in this section. Cactus0192837465 (talk) 23:07, 27 December 2018 (UTC)

Please, fellow Wpns, ... respect and countenance. Let's avoid words like disaster and certain symptoms of ownership.
 * I think that up to now the efforts on introducing the term "histogram" are at a suboptimal level, perhaps even at a misleading one.
 * I think it's "her", because it's really a person's weight, but I also think that having the referral after the pronoun sounds strange. (one could also have "his", but in talking about body weights it's "her" to me)
 * Presenting a histogram g as representing the function f might really be a misrepresentation in a section about "Representing a function", since I assume that neither Fourier-, nor Laplace-, nor Legendre-, nor ... transforms were originally targeted in this section as representing the original.
 * For the time being I consider histograms, impersonated by a function g, as transforms of functions f, and would prefer to remove the paragraph about histograms, leaving just the "bar graphs" as representing functions. Purgy (talk) 08:00, 28 December 2018 (UTC)
 * Histograms are produced out of a multiset, not a function. You take the multiset as input. Then, choose the bins, which is just a partition of the multiset in which elements of the same value must fall in the same partition but each partition can contain elements of different values. Usually the partitions are chosen such that each bin consists of values that are close together in some sense. Now count the number of elements of the multiset in each bin, and that gives you a function from the set of bins to the integers, or the reals (or rationals) if you compute frequency instead of number of elements. This function is the histogram of the multiset. The most common use is when the multiset is a sample. Functions are not the input. Cactus0192837465 (talk) 11:58, 28 December 2018 (UTC)


 * I'm inclined to agree that we should just nix this section. A histogram is fine when you have a bunch of observations and want to display the frequency that certain ranges occur, but trying to use that to visualize a function doesn't make any sense.  –Deacon Vorbis (carbon &bull; videos) 14:06, 28 December 2018 (UTC)

Paragraph about 'effective' drawing.
The paragraph

It is possible to draw effectively the graph of a function only if the function is sufficiently regular, that is, either if the function is differentiable (or piecewise differentiable) or if its domain may be identified with the integers or a subset of the integers.

is a popular misconception. The main part that is wrong with it is that 'effective' is nowhere defined in the paragraph. Then it comes the conclusion, which for most natural definitions of 'effective' doesn't follow. Differentiable or worse piecewise differentiable functions is (while not as wild as continuous) a very rich class functions. The reason is that they satisfy Whitney extension theorem, which can be used to break the mould for any sensible definition of what 'effective drawing' might be.

In order to have a correct paragraph with the same spirit, it is better to mention actual results like Kempe's universality theorem, which does provide a concrete possible notion of what 'effective' might mean, and once it was properly proven provides a precise class of graphs that can be drawn.

This article (functions) is quite basic. As a concequence lots of people feel that they can help adding information to it. Therefore, it is more likely that people who don't really have the proper understanding of what they are saying add claims to it. Some of the claims kind of sound right but aren't. This is one of them. — Preceding unsigned comment added by Cactus0192837465 (talk • contribs) 02:03, 29 December 2018 (UTC)
 * Why do you require a definition of "draw effectively"? This is common language that everybody can understand. The use of "effective" as a synonymous of "algorithmically computable" is clearly not implied here (this is the only mathematical meaning of "effective" that is commonly accepted by mathematicians). Also, the sentence that you quote does not asserts that every graph of a function can be drawn, but it asserts exactly the contrary, that is that if the function is not sufficiently regular, then the graph cannot be drawn (this is clear for a mathematician, but not necessarily for people that have never encounter functions whose graph cannot be drawn).
 * Yes the article is quite basic, and it is intended. Wikipedia is an encyclopedia. As such, the natural audience of this article is people who have only a vague notion of what mathematicians call a function. Therefore, technical results that would be confusing or not understandable for this audience should be avoided, when possible. The two results that you have linked above do not belong to the common knowledge of undergraduate students in mathematics, nor to the common knowledge of people who use functions in their professional work (physicists, engineers, ...) Thus, they do not belong to this article. Please read WP:TECHNICAL for more details on how a Wikipedia article should be written. D.Lazard (talk) 10:47, 29 December 2018 (UTC)
 * I see how you are reading the sentence. I understood the reverse implication because the subordinate sentences use 'if'. The 'only if' appears only in the content clause. I don't mean that a definition of effective is necesary in the article. But it is necessary if a claim of the form 'X implies effectively drawable' is made, as those subordinate clauses were saying. Take into account that even saying 'algorithmically computable' wouldn't work either because the nature of what are the basic operations allowed in the algorithm would need to be defined in the case of drawing. Different sets of basic operations yield different results. See drawing with straighedge and compas vs drawing with only straightedge, or drawing with different sets of origami axioms.
 * Whitney extension theorem of course has no place in this article. It belongs only to my explanation here in the talk page of why 'effectively drawable implies differentiable' cannot possibly be true.
 * On the other hand, mentioning Kempe's universality theorem has nothing technical to it. We are not proving it, only mentioning it. Its statement is quite basic. After all, the idea of drawing graphs that come from algebra using mechanical machines predates calculus by almost two millenia. Cactus0192837465 (talk) 14:57, 29 December 2018 (UTC)
 * You see, technical/non-technical and correct/incorrect are different categories. One can be non-technical while still making claims that when studied in more depth become true precise statements. Even the statement 'effectively drawable implies piecewise differentiable' could potentially be false. I don't know if someone could device a set of elementary operations so wild that differentiable function can be drawn.
 * For example, in the awful books of Stewart's calculus they present how to integrate some of the basic fractions skipping those with powers of irreducible quadratics. That's potentially fine for the sake of not being too technical. But then they claim that the content given allows to integrate the rational functions, which deceives the reader. The need for being non-technical doesn't demand necessarily claiming things that when dig into become falsehoods. Such things can most of the time be avoided. And it should, because students do take each instruction expand into them. Cactus0192837465 (talk) 15:31, 29 December 2018 (UTC)