Devil's staircase

A devil's staircase is a function f(x) defined on the interval [a,b] with the following properties:

• f(x) is continuous on [a,b].
• there exists a set N of measure 0 such that for all x outside of N the derivative f'(x) exists and is zero.
• f(x) is nondecreasing on [a,b].
• f(a) < f(b).

One staircase on [0,1] can be constructed as follows.

1. Express x in base 3.
2. Replace the first 1 with a 2 and everything after it with 0.
3. Replace all 2s with 1s.
4. Interpret the result as a binary number. The result is f(x).

This staircase is a cumulative distribution function; the random variable it describes is uniformly distributed on the Cantor set.

There are other functions that have been called "devil's staircase". One is defined in terms of the circle map[?].