Fabius function

In mathematics, the Fabius function is an example of an infinitely differentiable function that is not analytic anywhere, 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.

References

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

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