Talk:Cantor function

Talk items 2003 - 2005

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)

Is this what you mean? —Caesura 18:38, 10 Mar 2005 (UTC)

Wouldn't it be better to have the diagram be rendered in-page? It sounds possible, but I don't know how to do it.

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)

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

Question mark function

"The Minkowski question mark function visually loosely resembles the Cantor function, having the general appearance of a "smoothed out" Cantor function. The question mark function has the interesting property of having vanishing derivatives at all rational numbers, and yet being an absolutely continuous, strictly increasing function."

Retrieved from "http://en.wiki.x.io/wiki/Cantor_function"

The question mark function is not absolutely continuous, is it? It has a derivative of zero almost everywhere, and that coupled with absolute continuity seems to imply it is constant. Also, the article on the question mark function has no mention of its absolute continuity.

(oops, forgot to sign this one. Sorry) Polycrates 04:04, 28 September 2005 (UTC)

Well, its "absolutly continuous" using the standard high-school/college calculus definitions of continuity. If you know of a fancier definition that makes it not continuous, let me know. Also; I think you reveresed "almost everywhere" with "almost nowhere". The rationals are of measure zero, and so the question mark has zero derivatives only on a set of measure zero. It has infinite derivatives at the irrationals, and there are many, many, many more of those. (There are several ways of constructing its deriviative, which is very much not zero.) linas 23:11, 28 September 2005 (UTC)

Linas, I'm surprised. I'd have expected that you knew that absolute continuity is not the same as continuity. Lots of functions are continuous but not absolutely continuous. Michael Hardy 21:59, 29 September 2005 (UTC)

Ah, well, I don't know everthing. (I did know that, just informally). Anyway, the question mark is absolutely continuous. linas 00:09, 30 September 2005 (UTC)

Absolute continuity is a much stronger condition that simply being a Continuous function. Also, from the article on the question mark function:

"(the question mark function) has a derivative almost everywhere, of value zero"

I'm still not sure whether the information in this article is correct.

Polycrates 06:58, 29 September 2005 (UTC)

That article should not have said "almost everywhere". I'll fix it now. linas 00:09, 30 September 2005 (UTC)

It seems to me that everything but the picture of this artice is already included on the article about the Cantor set itself. I think that the Cantor function is way too closely related to the Cantor set to merit a separate page.

Yet another definition (Haussdorff measure)

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)

Moments

Sometimes this is called "Vitali's function" or the "Vitali-Cantor function". I don't know the history.

I have an answer to that. You mentioned an Italian paper! In some French books (I am French) it is called "Lebesgue function". Lebesgue and Vitali were both born in 1875. They were 9 years old when the function was described by Cantor (in 1884 I think). One can see it in Acta Mathematica and in the book about Letters from Cantor (in German: Cantor Briefe). The function was also described by the young Ludwig Scheeffer, who had met Cantor at that time. Perhaps he was the true inventor. Bdmy (talk) 09:14, 2 December 2008 (UTC)

Nice. And backing that up, from the Dovgoshey survey mentioned below : "This function was also used by H. Lebesgue in his famous 'Lecons sur l'integration et la recherche des fonctions primitives' (1904)". Cantor's paper, dated November 1883, has the function as a counterexample to Harnack: "De la puissance des ensembles parfaits de points", Acta. Math. 4 (1884) pp. 381-392. Reprinted in 1980. I'm still looking for Vitali. ~~Craig

So I have to correct my statement: Vitali and Lebesgue were 8 year old..

If you like history, there is a little inaccuracy in the paper by Dovgoshey and al. about the length of the Cantor curve. The following article: L. Scheeffer, Allgemeine Untersuchungen über Rectification der Curven, Acta Math. 5 (1884), 49–82, is precisely devoted to a study of length of graphs of functions, extending the classical differentiable setting, and one of the example is.. the reciprocal of the Cantor function. According to Hawkins quoting Cantor, it is Scheeffer who pointed out to him (to Cantor's great pleasure) that the paper by Harnack had a mistake. See the book by T. Hawkins, Lebesgue’s Theory of Integration ; its Origins and Development, Chelsea Publishing Company (1970). Bdmy (talk) 08:53, 4 December 2008 (UTC)

The moments can be calculated without much effort, using the self-similarity. I do not mean the moments of the probability distribution which has cumulative distribution equal to the Cantor function, which are already given in the article Cantor distribution. I mean the integrals of x^n*f(x). See: MR0698862 (84h:26012) Barbieri, Francesco; Zironi, Fernando An experimental study of singular functions: the moments of the Vitali function. (Italian) Atti Sem. Mat. Fis. Univ. Modena 30 (1981), no. 2, 264--283 (1983)

The use of these moments is in, e.g., approximating the function with polynomials.

Finally, there's the following great remark from http://www.cut-the-knot.org/do_you_know/cantor.shtml

"The Cantor function is as famous as it is useful for other exceptional constructs. For example, let g: [0, 1]->[0, 1]×[0, 1] be a Peano curve. Then g(f): [0, 1]->[0, 1]×[0, 1] is another Peano curve that is constant almost everywhere."

A recent survey of information on the Cantor function is: Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M. The Cantor function. Expo. Math. 24 (2006), no. 1, 1--37.

--Craig

Better Mathematical Explanation

I'm having difficulty understanding how a continuous function can have zero derivatives and infinite derivatives without any values in-between. I can envision a continuous function with zero derivatives everywhere except a small interval which includes a point with an infinite derivative, and I can envision a continuous relation with only zero and infinite derivatives that is not a function (think of a staircase), and I can envision a function with jump discontinuities, but I can't understand this. Is there any intuitive way of grasping this function? Eebster the Great (talk) 05:36, 15 December 2008 (UTC)

It has infinite derivatives at each member of the Cantor set, i.e. at those numbers between 0 and 1 whose ternary (base 3) expressions can be written with only 0s and 2s. Since that set is nowhere dense, between any two members of the Cantor set not only will there be others, but there will also be open intervals which are not in the Cantor set where the derivative is zero. That's the mathematics; my intuitive way of thinking about it is an infinitesimal and equal jump up at each member of the Cantor set. --Rumping (talk) 20:42, 14 July 2009 (UTC)

The Cantor function does not have infinite derivatives in the Cantor set (it has infinite upper right Dini derivative but not infinite upper left Dini derivative). The paper Dovgoshey, Mario, Ryazanov, and Vuorinen, The Cantor function, Expositiones Mathematicae, 24, 2006, pp. 1-37 explains that (see especially Theorem 8.3).

Integral

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)

Devil's staircase?

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)

It already does, in the lead. Baccyak4H (Yak!) 17:22, 30 April 2009 (UTC)

Graph length

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)

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 $k2^{-n}$, 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) --Bdmy (talk) 20:25, 3 June 2009 (UTC)

A recent version of the article states, "The Cantor function is non-decreasing, and so in particular its graph defines a rectifiable curve." I am uncomfortable with this statement: the function is non-decreasing *therefore* its graph is rectifiable. There's no mention of continuity, for starters; and even if included as a hypothesis, I'm not seeing this one as intuitively obvious. Can anyone offer a clearer statement? Thanks, 70.116.95.66 (talk) 01:18, 30 April 2013 (UTC)

Continuity?

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)

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)

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

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)

Explain the uniform convergence

Hello. Could someone explain how I can get to the second formula in the iterative part from the first one, to show that f_n converges uniformly. I mean the mathematical expression following "If ƒ denotes the limit function, it follows that, for every n ≥ 0,". 79.136.62.165 (talk) 10:16, 26 November 2012 (UTC)

Proof:

$\max _{x\in [0,1]}|f_{n+1}(x)-f_{n}(x)|\leq {\frac {1}{2}}\,\max _{x\in [0,1]}|f_{n}(x)-f_{n-1}(x)|,\quad n\geq 1.$

By induction,

$\max _{x\in [0,1]}|f_{n+1}(x)-f_{n}(x)|\leq 2^{-n}\max _{x\in [0,1]}|f_{1}(x)-f_{0}(x)|$

Hence,

${\begin{aligned}\max _{x\in [0,1]}|f(x)-f_{n}(x)|&=\max _{x\in [0,1]}\left|\sum _{k=n}^{\infty }(f_{k+1}(x)-f_{k}(x))\right|\\&\leq \max _{x\in [0,1]}\sum _{k=n}^{\infty }|f_{k+1}(x)-f_{k}(x)|\\&\leq \sum _{k=n}^{\infty }\max _{x\in [0,1]}|f_{k+1}(x)-f_{k}(x)|\\&\leq \sum _{k=n}^{\infty }2^{-k}\max _{x\in [0,1]}|f_{1}(x)-f_{0}(x)|\\&=2^{-n+1}\,\max _{x\in [0,1]}|f_{1}(x)-f_{0}(x)|\end{aligned}}$

— Arthur Rubin (talk) 11:11, 26 November 2012 (UTC)

Harvard references not working

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)