Talk:Absolute continuity

message: I am not too familiar with absolute continuity but it seems a little odd that it should be stronger than Lipschitz, as the lead paragraph currently claims.  Could someone add a clarification?  Katzmik (talk) 12:09, 27 October 2008 (UTC)
thread_title:  Lipschitz vs absolute 

message: Something strange happened to the definition in the last two edits. Reverted. Boris Tsirelson (talk) 08:27, 18 June 2009 (UTC)
thread_title:  Reverting 

message: Am I correct in assuming that this interval does not have to be finite?
TomyDuby (talk) 04:19, 22 November 2009 (UTC)
thread_title:  Interval ''I'' in the definition 

message: This should really be two articles: Absolutely continuous function and Absolutely continuous measure. Currently the article has two self-contained sections about two relatively distinct subjects, both of which would seem to be worthwhile subjects for articles of their own. There seems to be no reason for this other than inertia. The current referencing style makes in unclear which source corresponds to which subject, so the references would reviewed to see which article they should be placed in.--RDBury (talk) 20:11, 18 February 2010 (UTC)
replies: text: No, I believe, these two notions are very much related (as written in Sect.2.1). The sources could be better; I'll add more sources soon. And they will be sources that do describe both subjects (and their relation). True, the relation becomes weak when we turn to functions with values in metric spaces. This case could be treated in a separate section, for convenience of readers that are interested only in real-valued functions. Boris Tsirelson (talk) 20:11, 3 March 2010 (UTC)

text: Two books are added. I'll also add some inline citations (but not for metric spaces). Boris Tsirelson (talk) 20:28, 3 March 2010 (UTC)
thread_title:  Split 

message: Today, I made what should have been a very small (one character) edit to the bit on R-N theorem. I apologise that it took me 4 attempts to get mu and nu the correct way around in the text of my accompanying edit summary. Hence the multiple undos and redos whose sole purpose was to correct the edit summary, not the page itself.  — Preceding unsigned comment added by 138.40.68.40 (talk) 21:25, 7 March 2013 (UTC)
thread_title:  Explanation re a small edit on 7th March 

message: "(These examples are continuous but not uniformly continuous.)" — True. but somewhat misleading: a uniformly continuous function (even on [0,1]) need not be absolutely continuous (try the Cantor function). Boris Tsirelson (talk) 09:07, 3 July 2014 (UTC)
replies: text: Hi Boris. What you say is correct. I was tryin' to say that these are examples which are not absolutely continuous because they are not uniformly continuous, but that there are other functions (like the Cantor function) that are uniformly continuous but not absolutely continuous. But here's a question for you: What about the function x\sin(1/x) in the domain (0, 1]? Is it uniformly continuous? Absolutely continuous? Eric Kvaalen (talk) 12:41, 4 July 2014 (UTC)
replies: text: OK with "because". Now, the function x\sin(1/x) on the domain (0, 1] is not absolutely continuous; moreover, it is not of bounded variation. Try x_n=1/((n+0.5)\pi); you get the divergent harmonic series. And the derivative of this function is not integrable. On the other hand, it is uniformly continuous, since it can be extended by continuity to [0,1]. Also, the function x^{1+\varepsilon}\sin(1/x) on the domain (0, 1] is absolutely continuous whenever \varepsilon>0. Boris Tsirelson (talk) 16:55, 4 July 2014 (UTC)

text: And by the way, the "Equivalent definitions" are still equivalent on a bounded (but maybe not closed) interval. Boris Tsirelson (talk) 17:10, 4 July 2014 (UTC)
replies: text: Yeah. It's interestin' to think that no matter how small δ is, you can find an infinite amount of "variation" in the function x\sin(1/x). in the interval (0, δ)! Eric Kvaalen (talk) 06:15, 6 July 2014 (UTC)
replies: text: And no wonder. Given a divergent series, no matter how far is its tail, you can find an infinite sum in the tail. Boris Tsirelson (talk) 08:15, 6 July 2014 (UTC)
thread_title:  Absolute continuity of functions 

message: The explanation in the subsection "Relation between the two notions of absolute continuity" uses the term "locally absolutely continuous", but this term was not defined before. Can someone please add a definition? --Erel Segal (talk) 14:28, 22 August 2014 (UTC)
replies: text: Really, here "locally" is not needed, since the measure is assumed to be finite. (Otherwise, "locally" is needed, and means: on every bounded interval.) Boris Tsirelson (talk) 15:18, 22 August 2014 (UTC)
thread_title:  Undefined term: "locally absolutely continuous"