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In mathematics, the '''Fabius function''' is an example of an infinitely differentiable function that is nowhere [[analytic function|analytic]], found by {{harvtxt|Fabius|1966}}.
 
The Fabius function is defined on the unit interval, and is given by the [[probability distribution]] of
 
:<math>\sum_{n=1}^\infty2^{-n}\xi_n,</math>
 
where the ''&xi;''<sub>''n''</sub> are [[independence (probability)|independent]] [[uniform distribution (continuous)|uniformly distributed]] [[random variable]]s on the [[unit interval]].
 
==References==
{{Citation | last1=Fabius | first1=J. | title=A probabilistic example of a nowhere analytic C<sup></sup>-function | mr=0197656  | year=1966 | journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete | volume=5 | pages=173–174}}
 
[[Category:Types of functions]]
 
 
{{mathanalysis-stub}}

Revision as of 06:17, 20 April 2013

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Template:Harvtxt.

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

n=12nξn,

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

References

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