Talk:Pointwise convergence

[Untitled]

I'm not quited convinced that the series {$x^{n}$} is not uniformly convergent on the interval [0,1).The limit function f(x) is not continious at x₀=1. For every 0<x<1 the following is true: lim (n->inf) [sup {|f_n(x)-f(x)|} ]=0

No, that last assertion is false. The value of

$\sup\{\,|f_{n}(x)-f(x)|:0\leq x\leq 1\,\}$

is 1, for every n. That is because, for any fixed value of n, xⁿ can be made as close to 1 as desired by making x close enough, but not equal, to 1. So f_n(x) approximates 1 while the limiting function f(x) remains 0, as x approaches 1. Michael Hardy (talk) 15:39, 5 March 2008 (UTC)

whoa, I am pretty sure that is not true. For any value x < 1, as n goes to infinty, x^n goes to 0. So that function does converge uniformly. —Preceding unsigned comment added by 126.109.110.248 (talk) 12:15, 13 March 2011 (UTC)

Is the definition of uniform convergence in the section "Properties" true?

Hi.

In the section "Properties" of this article "Pointwise convergence", it is stated that a series of functions ${\textstyle \{f_{n}\}}$ uniformly converges to ${\textstyle f}$ if ${\textstyle \lim _{n\to \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0}$ is true. (Interpretation: if the maximum absolute difference between f_n and f is infinitesimally close to 0 as n goes to infinity, then it is said that f_n uniformly converges to f.). But is it true? More specifically, how it is equivalent to the uniform convergence definition in the uniform convergence main article "Uniform convergence"?

The definitions of the uniform convergence in both articles must be equivalent, so I need someone who can clarify this.

[Definition of the uniform convergence in the article "Uniform convergence"]

A sequence of functions $(f_{n})$ converges uniformly to a limiting function $f$ on a set $E$ as the function domain if, given any arbitrarily small positive number $\varepsilon$, a number $N$ can be found such that each of the functions $f_{N},f_{N+1},f_{N+2},\ldots$ differs from $f$ by no more than $\varepsilon$ at every point $x$ in $E$. Goodphy (talk) 10:57, 11 February 2026 (UTC)

Of course they are equivalent; this is a straightforward application of the definition of lim followed by a straightforward application of the definition of sup. The definition here is also present in the article Uniform_convergence#Definition, see the sentence beginning "In yet another equivalent ...". --JBL (talk) 20:08, 11 February 2026 (UTC)

Hi.

Thank you for your response. I'm still confusing if there is a case where ${\textstyle \forall x\in X,\lim _{n\to \infty }f_{n}(x)=f(x)}$ (pointwise convergence) does not guarantee ${\textstyle \lim _{n\to \infty }\,\sup\{\,\left|f_{n}(x)-f(x)\right|:x\in A\,\}=0}$ (uniform convergence).

If ${\textstyle f_{n}(x)}$ is infinitesimally close to ${\textstyle f(x)}$ at every ${\textstyle x}$ as ${\textstyle n\to \infty }$, then how the maximum (absolute) difference between ${\textstyle f_{n}(x)}$ and ${\textstyle f(x)}$ can't be infinitesimally close to zero as ${\textstyle n\to \infty }$? Goodphy (talk) 13:15, 1 March 2026 (UTC)

Please, explore your interest in mathematics by taking a class at a local university or reading a good-quality textbook or going through OCW or whatever, not by trying to edit technical encyclopedia articles on subjects you don't understand well. The statement ${\textstyle f_{n}(x)}$ is infinitesimally close to ${\textstyle f(x)}$ at every ${\textstyle x}$ as ${\textstyle n\to \infty }$ seems to be something you made up (at least, it does not appear in the article). It is informal; the natural way to formalize it (making the appropriate quantifiers explicit) is as the definition of uniform convergence, not as the definition of pointwise convergence. The difference between these two concepts is discussed at length in the article Uniform convergence and in this article; in the present article, there are several examples, including the simple example $f_{n}(x)=x^{n}$ on the domain $[0,1]$. It is not the role of Wikipedia talk-pages to provide you individualized tutoring. --JBL (talk) 00:20, 2 March 2026 (UTC)

Hi.

I have raised a question in this article's TALK page because I thought there could be incorrect / insufficient / or inconsistent information in this Article, instead of editing the article. I avoid editing article if I'm not sufficiently sure, and using the TALKS page is one of ways to avoid it.

If you don't want to teach me, don't do. I have no intension to use you as a tutor. Goodphy (talk) 01:45, 2 March 2026 (UTC)

Well, the article could certainly use more references. It is unfortunately hard to search for formulas like this in the search engines available, so inline citations to textbooks are often not present. I would think these properties would show up in the textbooks in the references section, but nobody has looked up the page numbers and I would honestly not be surprised if some of these ended up as exercises - which are, of course, notorious for having errors and misstatements compared to the main body of the textbook. So all this is to say that there is a lot of work to do on Wikipedia and relatively little time to do it. So if you want to improve this article then I would say just be wp:bold and find the references. Mathnerd314159 (talk) 07:09, 3 March 2026 (UTC)

Hi. @Mathnerd314159.

Thank you to motivate me to find a source of the uniform convergence definition in this article, and I found it, so I added an inline citation to specifically mention the source (book pages).

The contents of the source pages are essentially same to one mentioned in the Definition section of the Wikipedia article "Uniform convergence", as mentioned by @JayBeeEll above, but reading the source book made me understand this definition somehow. Goodphy (talk) 12:43, 8 March 2026 (UTC)