In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by. It was also written down as the Fourier transform of by. The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of where the ξn are independent uniformly distributed random variables on the unit interval. This function satisfies the initial condition f(0) = 0, the symmetry condition for and the functional differential equation for It follows that f(x) is monotone increasing for with f(1/2)=1/2 and f(1)=1. There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f(x) = 0 for x ≤ 0 , f(x + 1) = 1 − f(x) for 0 ≤ x ≤ 1 , and f(x + 2r) = −f(x) for 0 ≤ x ≤ 2r with r a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.

Values

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments.