Editing Cantor function (section)

== Generalizations ==
Let

: <math>y=\sum_{k=1}^\infty b_k 2^{-k}</math>

be the [[dyadic rational|dyadic]] (binary) expansion of the real number 0 ≤ ''y'' ≤ 1 in terms of binary digits ''b''<sub>''k''</sub> &isin; {0,1}. This expansion is discussed in greater detail in the article on the [[dyadic transformation]]. Then consider the function

: <math>C_z(y)=\sum_{k=1}^\infty b_k z^{k}.</math>

For ''z''&nbsp;=&nbsp;1/3, the inverse of the function ''x'' = 2&nbsp;''C''<sub>1/3</sub>(''y'') is the Cantor function.  That is, ''y''&nbsp;=&nbsp;''y''(''x'') is the Cantor function. In general, for any ''z''&nbsp;&lt;&nbsp;1/2, ''C''<sub>''z''</sub>(''y'') looks like the Cantor function turned on its side, with the width of the steps getting wider as ''z'' approaches zero.

As mentioned above, the Cantor function is also the cumulative distribution function of a measure on the Cantor set. Different Cantor functions, or Devil's Staircases, can be obtained by considering different atom-less probability measures supported on the Cantor set or other fractals. While the Cantor function has derivative 0 almost everywhere, current research focuses on the question of the size of the set of points where the upper right derivative is distinct from the lower right derivative, causing the derivative to not exist. This analysis of differentiability is usually given in terms of [[fractal dimension]], with the Hausdorff dimension the most popular choice. This line of research was started in the 1990s by Darst,<ref>{{Cite journal|title = The Hausdorff Dimension of the Nondifferentiability Set of the Cantor Function is [ ln(2)/ln(3) ]2|jstor = 2159830|journal = [[Proceedings of the American Mathematical Society]]|date = 1993-09-01|pages = 105–108|volume = 119|issue = 1|doi = 10.2307/2159830|first = Richard|last = Darst}}</ref> who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set, <math>(\log_3(2))^2</math>. Subsequently [[Kenneth Falconer (mathematician)|Falconer]]<ref>{{Cite journal|title = One-sided multifractal analysis and points of non-differentiability of devil's staircases|journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]| date = 2004-01-01|issn = 1469-8064|pages = 167–174|volume = 136|issue = 1|doi = 10.1017/S0305004103006960|first = Kenneth J.|last = Falconer|authorlink=Kenneth Falconer (mathematician)| doi-broken-date=1 November 2024 |bibcode = 2004MPCPS.136..167F|s2cid = 122381614}}</ref> showed that this squaring relationship holds for all Ahlfors's regular, singular measures, i.e.<math display="block">\dim_H\left\{x : f'(x)=\lim_{h\to0^+}\frac{\mu([x,x+h])}{h}\text{ does not exist}\right\}=\left(\dim_H\operatorname{supp}(\mu)\right)^2</math>Later, Troscheit<ref>{{Cite journal|title = Hölder differentiability of self-conformal devil's staircases|journal = [[Mathematical Proceedings of the Cambridge Philosophical Society]]|date = 2014-03-01|issn = 1469-8064|pages = 295–311|volume = 156|issue = 2|doi = 10.1017/S0305004113000698|first = Sascha|last = Troscheit|arxiv = 1301.1286|bibcode = 2014MPCPS.156..295T|s2cid = 56402751}}</ref> obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and [[Self-similarity|self-similar sets]].

[[Hermann Minkowski]]'s [[Minkowski's question mark function|question mark function]] loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion.  The question mark function has the interesting property of having vanishing derivatives at all rational numbers.