The Fabius function is defined on the unit interval, and is given by the probability distribution of
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+2r) = −f(x) for 0≤x≤2r with r≥1 integer; it is strongly related to the Thue–Morse sequence.
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