Talk:Cantor function

message: Can someone figure out how to adjust the size of the illustration in this article?  I've played with it a bit without success.  Michael Hardy 14:07, 29 Sep 2003 (UTC)
replies: text: Is this what you mean? —Caesura 18:38, 10 Mar 2005 (UTC)
replies: text: Thanks; I've edited the article in a manner similar to what you did above.  I don't know what you mean by "rendered in-page", though. Michael Hardy 02:46, 11 Mar 2005 (UTC)

text: Could you explain what you mean by "rendered in-page"? —Caesura 18:38, 10 Mar 2005 (UTC)
thread_title:  Talk items 2003 - 2005 

message: I have added another def for the Cantor function,
stresses the meaning of the picture:
The Cantor function equals the measure of sections
of the Cantor set.
I did not worry about the special role of rational arguments
in the definition since they have zero measure anyway.
--Benjamin.friedrich 16:10, 10 January 2007 (UTC)
thread_title:  Yet another definition (Haussdorff measure) 

message: Since the function is continuous everywhere, by The Fundamental Theorem of Calculus it should have a definite integral over every interval.  How can we find these?  I'm especially having difficulty envisioning a sensible partition.  Eebster the Great (talk) 05:40, 15 December 2008 (UTC)
thread_title:  Integral 

message: Would it be appropriate to mention in the article that the function is also sometimes known as the "Devil's staircase"?  128.97.19.143 (talk) 17:16, 30 April 2009 (UTC)
replies: text: It already does, in the lead. Baccyak4H (Yak!) 17:22, 30 April 2009 (UTC)
thread_title:  Devil's staircase? 

message: An anonymous editor has been adding the unjustified statement that the length of the graph of the Cantor function is 1, less than the \sqrt 2 distance between the endpoints.  Perhaps something could be said to the effect that the length of the graph is 2, but I don't know whether it's interesting, nor do I have a source available.  — Arthur Rubin  (talk) 14:14, 3 June 2009 (UTC)
replies: text: That the length is indeed equal to 2 is proved, among more general facts,  in a 1884 paper in Acta Mathematica by Ludwig Scheeffer, Allgemeine Untersuchungen ûber Rectification der Curven, Acta Math. 5 (1884), 49–82.  Actually Scheeffer started with the function reciprocal to the Cantor function: a pure jump function with jumps at a countable dense subset of [0,1], but he also introduced the reciprocal (the Cantor function of this article is a special case, for special choice of dense subset —the dyadic numbers— and special jump values, 3^{-n} at all dyadic numbers of the form k 2^{-n}, with k an odd integer.  Scheeffer also observed that the "Cantor function" (that was not yet published by Cantor) is continuous, has 0 derivative outside a closed set with 0 content (today: measure 0), and yet is not constant.  Cantor's function is perhaps Scheeffer's function... or at least Cantor-Scheeffer function.  See also the mentions to Scheeffer in T. Hawkins, Lebesgue’s Theory of Integration ; its Origins and Development, Chelsea Publishing Company (1970). --Bdmy (talk) 18:50, 3 June 2009 (UTC)
thread_title:  Graph length 

message: The function is continuous, but the article does nothing to convince the reader of this fact, and intuition certainly fails where that graph is concerned.  Perhaps there should be a section of the proof?  Twin Bird (talk) 18:31, 1 October 2009 (UTC)
replies: text: I edited one of the sections to point out that the non-atomic nature of the measure in question directly corresponds to the continuity of the corresponding cumulative distribution function. — Carl (CBM · talk) 20:50, 1 October 2009 (UTC)

text: I find the graph perfectly convincing w.r.t continuity 87.77.17.135 (talk) 13:55, 15 June 2011 (UTC)

text: Proof: the function maps [0, 1] to itself and it is increasing. Since its range is the whole [0, 1] it cannot have discontinuities.  — Preceding unsigned comment added by Mbtnt (talk • contribs) 14:04, 17 December 2020 (UTC)
thread_title:  Continuity? 

message: The first two Harvard references not working (the two in the intro to 1884).  That is, clicking on "1884" doesn't take you to the reference. The next two are working.  I don't know how to fix them.  Bubba73 You talkin' to me? 04:26, 22 May 2019 (UTC)
thread_title:  Harvard references not working