Fabius function

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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


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


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