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.