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 $0.a_1 a_2 a_3 \ldots$ , then let $N$ be $\infty$ 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 $n < N$ and let $b_N = 1$ . We define the Cantor function (or the Cantor ternary function) as \begin{equation*} f(x) = \sum_{n=1}^N \frac{b_n}{2^n}. \end{equation*} This function can be easily checked to be continuous and monotonic on $[0,1]$ and also $f'(x) = 0$ 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.

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.

Bibliography

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H. L. Royden. Real Analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1988