Devil's staircase

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Image:Devils-staircase.svg 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.

Another example is the question mark function.

If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can be 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 an absolutely continuous random variable (since the probability density is zero everywhere it exists).

The Devil's staircase occurs naturally in some dynamical systems. Most famously, perhaps, it is at the center of the fractional quantum Hall effect.