View source for Cantor ternary function

{{MSC|26A45|28A15}}

[[Category:Analysis]]

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The Cantor ternary function (also called Devil's staircase and, rarely, Lebesgue's singular function) is a continuous monotone function $f$ mapping the interval $[0,1]$ onto itself, with the remarkable property that its derivative vanishes almost everywhere (recall that any monotone function is differentiable almost everywhere, see for instance [[Function of bounded variation]]). 

The function can be defined in the following way (see, for example, Exercise 46 in Chapter 2 of {{Cite|Ro}}). Given $x\in [0,1]$ consider its ternary expansion $\{a_i\}$, i.e. a choice of coefficients $a_i\in \{0,1,2\}$ such that
\[
x = \sum_{i=0}^\infty \frac{a_i}{3^i}\, .
\]
Define $n(x)$ to be 
* $\infty$ if none of the coefficients $a_i$ takes the value $1$
* the smallest integer $n$ such that $a_n=1$ otherwise.
Then
\[
f(x) = \sum_{i=0}^{n(x)-1} \frac{a_i}{2^{i+1}} + \frac{1}{2^{n(x)}}.
\]
For alternative definitions see Example 1.67 of {{Cite|AFP}} and page 55 of {{Cite|Co}}.

It follows trivially from the definition that $f$ is locally constant on the complement of the [[Cantor set]] $C$: since the Cantor set is a set of Lebesgue measure zero, the derivative of $f$ vanishes almost everywhere. For the same reason, the [[Generalized derivative|distributional derivative]] of $f$ is a [[Singular measures|singular measure]] $\mu$ supported on $C$. 

The Cantor function is a prototype of a [[Singular function|singular function]] in the sense of Lebesgue, cf. [[Lebesgue decomposition]]. The observation that $f$ is a function of bounded variation for which 
\[
f (1)-f(0) \neq \int_0^a f'(t)\, dt
\]
(where $f'$ denotes the classical pointwise derivative) was first made by Vitali in {{Cite|Vi}}. For this reason some authors use the terminology ''Cantor-Vitali function'' (see {{Cite|AFP}}).

==References==
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|valign="top"|{{Ref|AFP}}||     L. Ambrosio, N.  Fusco, D.  Pallara,  "Functions of bounded  variations   and  free  discontinuity  problems".  Oxford Mathematical  Monographs. The     Clarendon Press,  Oxford  University Press, New York,  2000.      {{MR|1857292}}{{ZBL|0957.49001}}  
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|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}
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|valign="top"|{{Ref|Vi}}|| A. Vitali, "Sulle  funzioni integrali",  Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur.,  '''40''' 1905 pp.  1021-1034.
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