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).

A standard example of a devil's staircase is the Cantor function, which is sometimes called "the" devil's staircase. There are, however, other functions that have been given that name. One is defined in terms of the circle map.