In mathematics, a devil's staircase is any function f(x) defined on the interval [a, b] that has 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 .
If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor a continuous random variable (since the probability density is zero everywhere it exists).