Talk:Cantor function

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[edit] 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)

[edit] 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.wikipedia.org/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.

[edit] 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)