Fabius function

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 ${\displaystyle f(0)=0}$, the symmetry condition ${\displaystyle f(1-x)=1-f(x)}$ for ${\displaystyle 0\leq x\leq 1,}$ and the functional differential equation

${\displaystyle f'(x)=2f(2x)}$

for ${\displaystyle 0\leq x\leq 1/2.}$ It follows that ${\displaystyle f(x)}$ is monotone increasing for ${\displaystyle 0\leq x\leq 1,}$ with ${\displaystyle f(1/2)=1/2}$ and ${\displaystyle f(1)=1}$ and ${\displaystyle f'(1-x)=f'(x)}$ and ${\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2.}$

It was also written down as the Fourier transform of

${\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^{m}}}\right)^{m}}$

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

${\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}$

where the ξ_n are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of ${\displaystyle {\tfrac {1}{2}}}$ and a variance of ${\displaystyle {\tfrac {1}{36}}}$.

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 + 2^r) = −f (x) for 0 ≤ x ≤ 2^r 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 Rvachev up function is closely related: up(x) = F(1 - |x|) for |x| ≤ 1.