Talk:Uniform convergence

message: In the section of theorems relating to the interchanging of limits, specifically the one pretaining to derivatives, I don't believe that uniform convergence of the original sequence of functions is necessary.  It should be that if the sequence of functions {f_n(x)} converges pointwise and the sequence of derivatives {f_n'(x)} of the original sequence converges uniformly, then the original limit function f(x) = lim f_n(x) is differentiable and its derivative is equal to the limit of the sequence of the derivatives of the original sequence, f'(x) = lim f_n'(x). --anon
: You are right. But I would prefer to leave it the way it is, it is more straightforward that way.  
: By the way, if the interval on which f_n are defined is bounded, it is enough for the sequence {f_n(x)} to be convergent just at one point. This together with the assumption that {f_n'(x)} converges uniformly, will imply that f_n converges uniformly to a function f and that f'(x) = lim f_n'(x). Oleg Alexandrov 02:37, 14 September 2005 (UTC)
thread_title: Theorem assumption too strong?

message: To someone with some graphics expertise: this article would benefit from an image of a uniform ε-neighborhood of a given function.  (Just a "tube" of diameter 2ε around the graph of a function.)  I would do it myself, but I'm pretty hopeless with graphics.  Sławomir Biały (talk) 12:32, 18 October 2010 (UTC)
thread_title:  Image request 

message: I wanted to add a bit to the history section to give a few more details.  Also I want to rework it to remove weasel words such as "Some historians", but I thought I would state my intention on the talk page first. Thenub314 (talk) 00:44, 17 December 2011 (UTC)
thread_title:  History Section

message: The second sentence in this Wikipedia article says "A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x."  I was under the impression that the speed of convergence may depend on x but is bounded, i.e. f doesn't converge arbitrarily quicker at certain points than at others.  Can someone confirm this?  And if so that statement should probably be changed.  — Preceding unsigned comment added by Zaubertrank (talk • contribs) 04:16, 3 March 2012 (UTC)
replies: text: "Speed of convergence" is a questionable concept. Mathematical things don't really have "speeds", even though they can be used to describe speeds in physics.98.67.106.59 (talk) 20:11, 4 August 2012 (UTC)
replies: text: No, speed of convergence is a valid concept (though it is better to say "rate", not "speed"). But indeed, the phrase "speed...does not depend on x" is somewhat misleading. On the other hand, "rate...is bounded" is hardly clear enough. Boris Tsirelson (talk) 20:19, 24 April 2014 (UTC)

text: Maybe this equivalent formulation could help: |f_n(x_n)-f(x_n)|\to0 for every sequence (x_n)_n. Boris Tsirelson (talk) 20:30, 24 April 2014 (UTC)
thread_title:  Incorrect Statement in the Introductory Paragraph 

message: My textbook on complex analysis states that if a sequence of analytic functions converges uniformly on a region of the complex plane, then the limit of the sequence is analytic in that region. This seems an important result (it shows that complex-differentiable functions are better behaved than real-differentiable functions) yet this article doesn't mention it at all. Perhaps it should be added under a new heading called "To analycity"?  — Preceding unsigned comment added by JacekW (talk • contribs) 09:05, 27 January 2013 (UTC)
thread_title:  Uniform convergence of analytic functions in the complex plane 

message: 1. Applications -> To continuity: is f continuous on S or I?
2. Proof: "Since fn is continuous at x0..." - which n is meant? If it is an arbitrary n, then \delta(\varepsilon) is rather \delta(n,\varepsilon).
Boris Tsirelson (talk) 06:03, 26 July 2015 (UTC)
thread_title:  Recent edits by [[User_talk:SergeyLiflandsky|SergeyLiflandsky]] 

message: See WT:MATH#Proofs, revisited  — Arthur Rubin  (talk) 17:58, 29 September 2015 (UTC)
replies: text: However, the discussion of whether the proof is correct and placed in (a) correct section is not really discussed there.  — Arthur Rubin  (talk) 18:03, 29 September 2015 (UTC)
thread_title:  Centralized discussion on proofs 

message: There is some controversy on whether Cauchy actually made a mistake here or not. See the following for some lively discussion: one, two, three, four. Double sharp (talk) 08:56, 22 May 2020 (UTC)
thread_title:  "Cauchy's wrong theorem"