Cantor function

The Cantor function is a canonical example of a singular function. It is based on the Cantor set, and is usually defined as follows. Let $x$ be a real number in $[0,1]$ with the ternary expansion $0.a_1{a}_2{a}_3\mathrm{\dots }$, then let $N$ be $\mathrm{\infty }$ if no $a_n=1$ and otherwise let $N$ be the smallest value such that $a_n=1$. Next let $b_n=\frac{1}{2}{a}_n$ for all and let $b_N=1$. We define the Cantor function (or the Cantor ternary function) as

$$f(x)=\sum _{n=1}^N\frac{b_n}{2^n}.$$

This function can be easily checked to be continuous and monotonic on $[0,1]$ and also $f^{\prime }(x)=0$ almost everywhere (it is constant on the complement of the Cantor set), with $f(0)=0$ and $f(1)=1$. Another interesting fact about this function is that the arclength of the graph is 2, hence the calculus arclength formula does not work in this case.

Figure 1: Graph of the cantor function using 20 iterations.

This function, and functions similar to it are frequently called the Devil’s staircase. Such functions sometimes occur naturally in various areas of mathematics and mathematical physics and are not just a pathological oddity.

Title	Cantor function
Canonical name	CantorFunction
Date of creation	2013-03-22 14:08:23
Last modified on	2013-03-22 14:08:23
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	9
Author	jirka (4157)
Entry type	Definition
Classification	msc 26A30
Synonym	Cantor ternary function
Synonym	Cantor-Lebesgue function
Synonym	Devil’s staircase
Related topic	CantorSet
Related topic	SingularFunction
Defines	Cantor function
Defines	Cantor ternary function