Fabius function
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 are independent uniformly distributed random variables on the unit interval.
This function satisfies the initial condition, the symmetry condition for and the functional differential equation for It follows that is monotone increasing for with and
There is a unique extension of to the real numbers that satisfies the same equation. This extension can be defined by for, for, and for with 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.
