Singular function
A devil's staircase is a function f(x) defined on [a,b] with the following properties:
- f(x) is continuous on [a,b].
- f'(x) is zero on [a,b] except at a set of measure 0.
- f(x) is nondecreasing on [a,b].
- f(a) <> f(b).
One staircase is computed as follows.
- Express x (which is in [0,1]) in base 3.
- Replace the first 1 with a 2 and everything after it with 0.
- Replace all 2s with 1s.
- Interpret the result as a binary number. The result is f(x).
This staircase is a probability distribution function; the random variable it describes is uniformly distributed on a Cantor set.
There are other functions that have been called "devil's staircase". One is defined in terms of the circle map.