Fabius function
Nowhere analytic, infinitely differentiable function
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In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).
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 and and .
It was also written down as the Fourier transform of
by Børge Jessen and Aurel Wintner (1935).
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. That distribution has an expectation of and a variance of .
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.
The Rvachëv up function[1] is closely related to the Fabius function f: It fulfills the delay differential equation[2] (See Delay differential equation for another example.)
