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[[File:Graph of the Fabius function.png|thumb|Extension of the function to the nonnegative real numbers.]]
In mathematics, the '''Fabius function''' is an example of an [[smoothness|infinitely differentiable function]] that is nowhere [[analytic function|analytic]], found by {{harvs|last=Fabius|first=Jaap|year=1966|txt}}. It was also written down as the Fourier transform of
:<math> \hat{f}(z) = \prod_{m=1}^\infty \left(\cos\frac{\pi z}{2^m}\right)^m</math>
by {{harvs|last1=Jessen|first1=Børge|and|last2=Wintner|first2=Aurel|year=1935|txt}}.
The Fabius function is defined on the unit interval, and is given by the [[probability distribution]] of
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==References==
*{{Citation | last1=Fabius | first1=J. | title=A probabilistic example of a nowhere analytic {{math|''C''{{hsp}}<sup>∞</sup>}}-function | mr=0197656 | year=1966 | journal=Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete | volume=5 | pages=173–174 | doi=10.1007/bf00536652}}
*{{Citation | last1=Jessen | first1=Børge | last2=Wintner|first2=Aurel| title=Distribution functions and the Riemann zeta function | mr=1501802 | year=1935 | journal=Trans. Amer. Math. Soc. | volume=38 | pages=48-88 | doi=10.1090/S0002-9947-1935-1501802-5 }}
*{{cite thesis|first1=Youri |last1=Dimitrov |title=Polynomially-divided solutions of bipartite self-differential functional equations
|year= 2006 |url= http://rave.ohiolink.edu/etdc/view?acc_num=osu1155149204}}
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