Fabius function

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

Extension of the function on the nonnegative real axis.

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

The Fabius function is defined on the unit interval, and is given by the probability distribution of

\sum_{n=1}^\infty2^{-n}\xi_n,

where the ξ_n are independent uniformly distributed random variables on the unit interval.

This function satisfies the functional equation f′(x)=2f(2x) (where f′ denotes the derivative of f) for 0≤x≤1. There is a unique extension of f to the nonnegative real numbers which satisfies the same equation: it can be defined by f(x+1) = 1−f(x) for 0≤x≤1 and f(x+2^r) = −f(x) for 0≤x≤2^r with r≥1 integer; it is strongly related to the Thue–Morse sequence.

References

Fabius, J. (1966), "A probabilistic example of a nowhere analytic C^∞-function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 5: 173–174, doi:10.1007/bf00536652, MR 0197656