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) for 0 ≤ x ≤ 1/2; here f ′ denotes the derivative of f. There is a unique extension of f to the nonnegative real numbers which satisfies the same equation. This extension 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 a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the Thue–Morse sequence.
- 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
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