Fabius function

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).

Graph of the Fabius function on the interval [0,1].

Extension of the function to the nonnegative real numbers.

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 Rvachëv up function^[1] is closely related: $${\displaystyle u(t)={\begin{cases}F(t+1),\quad |t|<1\\0,\quad |t|\geq 1\end{cases}}}$$ which fulfills the Delay differential equation^[2] $${\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).}$$ (see Another example).

Values

The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:^[3][4]

• ${\displaystyle f(1)=1}$
• ${\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}$
• ${\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}$
• ${\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}$
• ${\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}$
• ${\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}$
• ${\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}$
• ${\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}$

with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757.