Talk:Semi-continuity

Description inconsistent with formal definition and figures swapped
As on today, the opening of the article describes the semi-continuities as below. In fact, are not much higher (respectively, lower) should be replaced by are not much lower (respectively, higher). Also, as on today the graphs of lower semi-continuous and upper semi-continuous functions are swapped. The figure showing the upper semi-continuous function at $$x_0$$ is lower semi-continuous and vice varsa.

In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function $$f$$ is upper (respectively, lower) semi-continuous at a point $$x_0$$ if, roughly speaking, the function values for arguments near $$x_0$$ are not much higher (respectively, lower) than $$f\left(x_0\right).$$ Dr. Dinesh Karia (Talk) (contribs) 13:05, 8 September 2021 (UTC)


 * I'll check with a math PhD friend tonight, but you're onto something. I think that actually the diagram and the definition in the Characterization section (which are   the same) are correct, and the definition in Formal Definition  and the Intro, which is different, is wrong. The definition in Formal Definition  and Intro doesn't seem to me to make sense. It requires that in  a neighborhood of a point x_0,    f(x) > f(x_o), if I read it right, which would mean that a function was upper semicontinuous only at isolated point where it dipped discontinuously and then jumped back up. editeur24 (talk) 17:15, 8 September 2021 (UTC)


 * I read it wrong, so my objection in the last paragraph is invalid. editeur24 (talk) 20:25, 13 September 2021 (UTC)

Equivalence of definitions
I'm not sure the formal definition given is quite correct. For instance, consider the function $$f : \mathbb{R}\to [-\infty,\infty]$$ given by $$ f(x) := -1/x^2 $$ for $$ x\neq 0 $$ and $$ f(x):=-\infty $$ for $$ x=0 $$. This is upper-semicontinuous at $$ 0 $$ according to the limsup definition, but for any $$ \epsilon>0 $$ we have $$ f(0)+\epsilon = -\infty $$ and on every neighbourhood of $$ 0 $$ the function takes some finite values, which would mean that $$ f $$ is not upper-semicontinuous at $$ 0 $$ according to the definition given. If the definition instead reads "$$ f $$ is upper-semicontinuous at $$ x_0 $$ if for every $$ a>f(x_0) $$ there is a neighbourhood $$ U $$ of $$ x_0 $$ such that $$ f(x) -\infty\}$$, otherwise it fails (as your example illustrates). Zfeinst (talk) 17:51, 21 June 2013 (UTC)

First-countability
I suppose that the equivalences described in the "Formal definition"-section are equivalences only, if X is first-contable, since continuity always implies sequence-continuity, but the other direction being true only in first-countable spaces. Cf. Munkers: Topology, p. 190ff, or cf. the German Wikipedia article on "Halbstetigkeit", where X is required to be a metric space, which implies first-countability. Luzern66 (talk) 14:58, 3 July 2011 (UTC)

This definition covers only R and not R^n
This definition only covers functions with $$f : X \rightarrow \mathbb{R}$$. The definition for $$f : X \rightarrow \mathbb{R}^n$$ is important for fixed point theorems. --Clausen 23:31, 26 May 2005 (UTC)


 * I would challenge you to add a section at the bottom talking about R^n. :) This might be a better option than rewriting all the article in the general R^n case. Will you take the challenge? :) Oleg Alexandrov 00:25, 27 May 2005 (UTC)

Lower Semi-continous Example
I couldn't make any sense of this:

Imagine that you are scanning a certain scenery with your eyes and record the distance to the viewed object at all times. This yields a lower semi-continuous function which in general is not upper semi-continuous (for instance if you focus on the edge of a table).

Would someone like to clarify it? Or else we could just use a conventional example like $$f(x)= \lceil x \rceil $$ Deepak 16:17, 31 March 2006 (UTC)
 * Ok then, I'm going to make the change. Deepak 16:52, 20 April 2006 (UTC)

Are we sure about the lower semi-continuous definition in terms of having a neighborhood around x_0 where all the x in the nbhd have f(x) > f(x_0) ? given that we are also saying that a continuous function is both upper and lower semi-continuous, then taking the identity function and fixing any point should give us a l.s.c. function where any neighborhood will have some x with f(x1) < f(x0) < f(x2) where nbhd is fixed around x0, x1, x2 are in the nbhd. Maybe just use the lim inf definition.

18:12, 8 October 2006 (UTC)chuck


 * I think the article is correct as it stands: the neighbourhood definition does not state that f(x) > f(x_0) in the neighbourhood, but rather states that f(x) > f(x_0) - ε, and this would fit OK with continuous functions such as the identity function. Madmath789 21:50, 8 October 2006 (UTC)

Order topology
Upper (resp. lower) semi-continuity is continuity with respect to the right (resp. left) order topology on R. Is there any point mentioning this here? Algebraist 12:14, 24 March 2008 (UTC)
 * Not sure. That seems to be pretty esoteric stuff, and semi-continuity is already complicated enough for most readers, even with the ordinary topology. Oleg Alexandrov (talk) 15:04, 24 March 2008 (UTC)
 * Fair enough. That saves me the effort of finding a source, anyway. Algebraist 18:32, 28 March 2008 (UTC)
 * Are you sure? the order topology of R is equivalent to the standard one. (Anonymous)
 * I have now added this in the definitions section. It's really equivalent to some of the other characterizations, and conceptually useful in some contexts. PatrickR2 (talk) 09:07, 17 November 2021 (UTC)

The figure may involve confusion
The figure may involve a confusion with cadlag functions, it should be completed to avoid this confusion —Preceding unsigned comment added by 92.141.19.110 (talk) 13:40, 22 April 2010 (UTC)

You show the same graph with the same point x_0 as being both upper and lower semi-continuous. That would mean the function in the picture is continuous, which isn't. Could you please clarify? —Preceding unsigned comment added by Serban00 (talk • contribs) 01:11, 2 December 2010 (UTC)


 * The functions in the two graphs are almost the same, but they're not identical. On one graph, the open point is at the top of the jump discontinuity, and on the other, it's at the bottom. The open point indicates that the function does not exist at that point for x0, but does exist at the neighboring points up to that open point on one side; the open point is therefore the limit of f(x) as x approaches x0 from that one side. Both functions are called semi-continuous because f(x) exists on the neighboring points of f(x0), but on only one side instead of both. Mktyscn (talk) 05:08, 4 December 2010 (UTC)

Merge discussion (closed convex function)
I think the page on closed convex functions should be merged with semi-continuity since a closed convex function is merely a lower semi-continuous and convex function. What does everyone else think? Zfeinst (talk) 04:08, 28 May 2012 (UTC)

I disagree, because convex functions can be allowed to take the values +infinity and -infinity, and this introduces complications your reduction cannot capture. Some authors will call a convex function closed if it coincides with the pointwise supremum of its affine minorants; others require instead that every sub-level set be closed. When the value -infinity is allowed, these criteria are different. Perhaps the page on closed convex functions could be refined, but not in the suggested manner. — Preceding unsigned comment added by 137.82.36.82 (talk) 18:04, 4 July 2012 (UTC)


 * I think adding material on convex functions to this article would just make it more confusing. There is also probably ample scope for a separate article on closed convex functions.  That article needs more explanations, references, and examples.  Once those are added, I think it will be clear that these are distinct topics.   Sławomir Biały  (talk) 14:30, 23 February 2013 (UTC)

Topology of almost everywhere convergence?
In the examples section, it says that


 * Let $$(X,\mu)$$ be a measure space and let $$L^+(X,\mu)$$ denote the set of positive measurable functions endowed with the topology of $$\mu$$-almost everywhere convergence. Then the integral, seen as an operator from $$L^+(X,\mu)$$ to $$[-\infty,+\infty]$$ is lower semi-continuous. This is just Fatou's lemma.

I like the example, but it is false and I don't know a neat way to fix it. It is known that the notion of almost everywhere convergence doesn't come from a topology (unlike the notion of pointwise (everywhere) convergence, which comes from the product topology). If we use the topology of pointwise convergence (which would require not identifying two functions that are almost everywhere equal, or more easily, restricting to continuous functions), then the statement is well-formed but false.

How is this possible? Fatou's lemma talks about sequences and it is known that the convergent sequences in a topological space do not characterize its topology completely. A basic neighborhood of a function $$f$$ in the topology of pointwise convergence is a set of functions which nearly coincide with $$f$$ in a certain (finite) set of points, and this information is not enough to say anything about the values of the integrals of the functions. So all we could say is that integration is sequentially lower continuous. Marcosaedro (talk) 19:42, 27 January 2013 (UTC)


 * I have a thought about how to possibly repair this. Replace the "topology" on the space by that of convergence in measure.  This comes from a metrizable topology, so there is no awkwardness in saying the "topology of convergence in measure".  Fatou's theorem also holds for sequences that converge in measure, with essentially the same proof.  (Indeed, Fatou follows straightforwardly from Lebesgue's monotone convergence theorem.  For monotone sequences of functions, there is no difference between convergence in measure and convergence a.e., ... QED).   Sławomir Biały  (talk) 22:07, 22 February 2013 (UTC)

Some point-wise limits of lower semicontinuous functions.
Can the 'Semicontinuity' article be improved by adding the following (what appears to be) basic fact?

Fact. Every lower semicontinuous function f can be obtained as a point-wise limit from above of a monotone sequence fn of finite-valued lower semicontinuous functions.

Is this fact trivial/false/unknown etc? Any examples/counter-examples? Any references? Aucross (talk) 03:59, 5 February 2015 (UTC)

Suggestion for slight improvement
The text says "Multiplying a positive upper semi-continuous function with a negative number turns it into a lower semi-continuous function." But there seems to be no need to ask that the upper semi-continuous function is positive. We can actually deduce this from two useful facts: 1) $$f$$ is lsc iff $$-f$$ is usc because $$-f(x) < \alpha$$ iff $$f(x) > -\alpha$$ 2) multiplying by a positive real number preserves lsc and usc.

Streamlining of the definition section
The Formal Definition section of the article has become unnecessarily WP:TECHNICAL and should be simplified. Browsing at the history of the article, it appears that most of the changes introducing these technicalities were added by user. While some of his changes improved readability (for example having separate subsections for upper and lower semicontinuity, and the reordering of some of the sections), the additional minutiae made the new article harder than it needs to be. If other people agree with this, I will do the change.

In this particular case, the pre-Mgkrupa version of the definition (https://en.wikipedia.org/w/index.php?title=Semi-continuity&oldid=959632505) was correct, complete and to the point (for every $$y>f(x_0)$$ there is a neighborhood $$U$$ such that ...). Mgkrupa added several paragraphs of considerations about the case of $$+\infty$$ and "separating into two cases", etc, which seems totally unnecessary. While there is nothing logically incorrect about these paragraphs, they don't add anything to the understanding of the concept, and more importantly, they are all logical consequences of the definition anyway. These minutiae dilute the impact of the article and make it unnecessarily technical. They also go against WP:NOTTEXTBOOK. Sometimes "less is more" :-) PatrickR2 (talk) 23:56, 12 November 2021 (UTC)
 * I removed some of that info. Let me know if you want more changes. And feel free to make the changes yourself. Mgkrupa 00:43, 13 November 2021 (UTC)
 * Thank you Mgkrupa, this is a lot more readable. I appreciate that change. PatrickR2 (talk) 00:47, 13 November 2021 (UTC)
 * Good to see a nice polite interchange like that. Good for you guys. editeur24 (talk) 02:49, 13 November 2021 (UTC)

Semi-continuity versus semicontinuity
I am curious why all instances of "semicontinous" without a hyphen have been changed to the version with a hyphen. All the (pure mathematics) topology and analysis books that I have (Willard, Engelking, Royden, Stromberg, etc) seem to use the version without a hyphen. Same thing in the SIAM Review article in the Bibliography section of the article, and in https://encyclopediaofmath.org/wiki/Semicontinuous_function.

Also a search in both https://math.stackexchange.com and https://mathoverflow.net shows "semicontinuous" to be more commonly used. In addition the following comparison in Google books between "lower semicontinuous" and "lower semi-continuous" shows the version without a hyphen is about five times more common in recent books:
 * https://books.google.com/ngrams/graph?content=lower+semi-continuous%2Clower+semicontinuous&year_start=1910&year_end=2019&corpus=26&smoothing=3&direct_url=t1%3B%2Clower%20semi%20-%20continuous%3B%2Cc0%3B.t1%3B%2Clower%20semicontinuous%3B%2Cc0#t1%3B%2Clower%20semi%20-%20continuous%3B%2Cc0%3B.t1%3B%2Clower%20semicontinuous%3B%2Cc0

The version without a hyphen is also easier to type. PatrickR2 (talk) 16:36, 15 November 2021 (UTC)


 * Same type of results for searches in https://arxiv.org/search/math. The version without hyphen is used about fives times more commonly in academic research in mathematics. PatrickR2 (talk) 16:52, 15 November 2021 (UTC)


 * Would it be acceptable to you if I change the article to use the version without the hyphen? PatrickR2 (talk) 23:50, 16 November 2021 (UTC)


 * How nice of you to ask! Yes, it looks like people have checked it out. I think I like it better without the hyphen too, actually, just from the point of view of omitting a needless character. I think I may have asked a math professor friend with strong opinions who liked the hyphen, but go ahead and take it out.editeur24 (talk) 00:52, 17 November 2021 (UTC)


 * Thanks. I'll make the edit. PatrickR2 (talk) 01:09, 17 November 2021 (UTC)

Semicontinuous function as limit of monotone sequence of continuous functions
The article says "Every lower semi-continuous function on a uniform space (e.g. a metric space) arises as the supremum of a sequence of continuous functions". This is true when the domain is a metric space. But I don't think it is correct for uniform spaces in general. A topological space is uniformizable exactly when it is completely regular. On the other hand, it was proved by Hing Tong in 1952 that a topological space has the property that any lower semicontinuous function is the limit of a monotonically increasing sequence of continuous functions exactly when the space is perfectly normal. See Engelking exercise 1.7.15(c) for a sketch of the proof, also reproduced in https://math.stackexchange.com/questions/1279763. Since completely regular spaces are not even normal in general, the result cannot be true for uniform spaces. PatrickR2 (talk) 17:27, 15 November 2021 (UTC)


 * I have now fixed this. PatrickR2 (talk) 17:49, 15 November 2021 (UTC)

equivalences in the section on lower semicontinuity
In my understanding of the section on lower semicontinuity, condition (1) is not equivalent with the others, for instance with (3) (analogously the section on upper semicontinuity). Take the one-dimensional example f(x)=0 if x>0 and f(x)= +infinity else. Then, f is lower semicontinuous (actually: continuous) at every point of its domain (positive reals), but the sublevel set with respect to the level 0 (or any level y larger than zero) equals its domain which is open but not closed. 62.141.176.2 (talk) 15:11, 24 November 2022 (UTC)


 * I assume that you want the function $$f$$ in your example to be defined on the space $$X=\R$$. That function is not lower semicontinuous at $$x=0$$. And of course it's not continuous at that point either.  Note that the domain is all of $$\R$$ and not just the set of positive reals. PatrickR2 (talk) 02:22, 26 November 2022 (UTC)