Talk:Bounded function

Sin z not bounded?

 * The function f:R → R defined by f (x)=sin x is b
 * P    ounded. The sine function is no longer :bounded if it is defined over the set of all complex numbers.


 * why isn't sinz bounded function? --anon

Well, one has the equality
 * $$ \sin (z) = \frac{1}{2i}\left(e^{iz}-e^{-iz}\right) $$

which follows from Euler's formula. If z=1000i, which is an imaginary number, one has
 * $$ \sin (1000i) = \frac{1}{2i}\left(e^{i1000i}-e^{-i1000i}\right) $$
 * $$           = \frac{1}{2i}\left(e^{-1000}-e^{1000}\right) .$$

Now, $$e^{-1000}$$ is small, but $$e^{1000}$$ is huge, so this adds up to a huge number. Does that make sence? Oleg Alexandrov 21:05, 3 September 2005 (UTC)

proof?
how can one prove whether a function is bounded or not?

If a function is continuous and has a maxima and a minima at finite values of f(x), then it is bounded. If it is not continuous then you will have to find ways to find the upper bound and lower bound. Should both be finite, it will remain bounded. Umang me (talk) 06:11, 30 January 2010 (UTC)

Is this bounded?
Say I have an f:R->R continuous and strictly monotonically increasing for (-2, 2) and stationary at {-2, 2}. Then I have g(x) = f(x), x belonging to (-2, 2). Is g(x) bounded? I ask because g(x) does not attain a maxima or a minima at all, but is restricted by f(-2) < g(x) < f(2), where f(-2) and f(2) are finite real numbers. Therefore, (according to me) g(x) is bounded but does not attain its extremes. Am I right? Umang me (talk) 06:11, 30 January 2010 (UTC)

"The" Bounded Metric
The definition "$$f: X \to (M,d)$$ is bounded if $$\exists m \in M, K \in \mathbb{R} \forall x \in X : d(f(x),m)<K$$" doesn't feel quite right.

Let $$(M,d)$$ be a metric space and define a new metric $$d_b$$ on $$M$$ by $$d_b(x,y) = \min\{d(x,y),1\}$$. Then for any set $$X$$ and any function $$f: X \to (M,d_b)$$ we have that $$f$$ is bounded. Since $$(M,d)$$ is always (uniformly? I haven't checked) homeomorphic to $$(M,d_b)$$ this definition seems monumentarily useless.

What book is the definition in the article from? And wouldn't a more useful definition be to say that a function into a metric space is bounded if its image is totally bounded? Infenwe (talk) 18:04, 10 October 2010 (UTC)

Article expansion?
All but the simplest notions in this article (i.e. the drawing with two "limit lines" is quite useful) seem pretty subtle and require expansion, and maybe even proof. For example, I'm really uncomfortable with the d(f(x), a) < M business. I added the phrase to define "d" because I firmly believe that each symbol in an article should be defined (at least to some degree) in the article; I shouldn't have to go fishing for its definition.

I'd fix the article myself but I don't have the knowledge (indeed, that's why I came to the article, to get the knowledge). In particular the measurement notions e.g. talks of vector "distance" and etc seem optional (see below re Lay's treatment). They should be pulled out of the lead and put into their own subsections and perhaps expanded upon, but only after the main topic (bounded function) is defined and treated adequately.

Here are some sources: an intuitive treatment beginning with a geometric/topological treatment, e.g. from Hardy and Wright 2000:31 "The boundary C of R [H & W define first what a "region" R is in terms of open region or interior in a plane] is the set of points which are limit points of R but do not themselves belong to R. Thus the boundry of a circle is its circumference. A closed region R* is an open region R together with its boundary." etc etc. But this is not good enough because it doesn't include the notion of "function". So. ..

So I go hunting and find another book (I see my kid's frantic scribbling in it, including "I don't get this!" in "Theorem: Every convergent sequence is bounded" ) that provides definitions in terms of topological notions as well, i.e.
 * Stephen R. Lay, 1986, Analysis with an Introduction to Proof, 2nd Edition, Prentice Hall, Upper Saddle River NJ, ISBN:0-13-033267-4.

It takes him 88 pages and 3 sections to get to "bounded function" on page 168. He proceeds through sections 3 The real numbers, 4 Sequences, and 5 Limits and Continuity to cover in depth the knowledge necessary to understand this article. Here's his approach: Rather than begin with a full-blown set theory Section 3 assumes "an arithmetic" but also a "well ordering axiom" together with the notion of "induction" leading to "ordered fields" and a "completeness axiom". These lead to stuff that looks like it came out of Hardy and Wright, re rational functions, then notions of "boundary" and "interior" that in turn lead to "Compact sets" and the "Heine-Borel theorem"; section 3 ends finally at the optional "metric spaces". Then section 4 Sequences builds on section 3 to define convergence of a sequence, boundedness of a sequence etc. In chapter 5 I find "limits of functions", "one-sided limits", "continuous functions" (and discontinuous functions), at last arriving at "bounded function" on page 168 (this requiring understanding of the Heine-Borel theorem ), and aends with the optional section on "continuity in metric spaces".

So, anyway, the above 2 sources seem to offer an approach to further development of this article. At least for me the article is totally inadequate, both as a source for further reading, and as a source of knowledge. At least I can plug one hole by adding the Lay book as a reference. Bill Wvbailey (talk) 18:57, 31 March 2011 (UTC)

"Unbounded" listed at Redirects for discussion
A discussion is taking place to address the redirect Unbounded. The discussion will occur at Redirects for discussion/Log/2021 February 25 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 21:39, 25 February 2021 (UTC)

Image description
I propose changing the description of the image on top at the right, to indicate that the bounded function is shown by the pink or magenta arrow, rather than the red arrow, which I find to be a more precise definition. 2001:A61:25C1:C501:382F:BA3D:5595:9343 (talk) 11:41, 6 May 2023 (UTC)