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There is a unique extension of {{mvar|f}} to the nonnegative real numbers which satisfies the same equation. This extension can be defined by {{math|1=''f''{{hsp}}(''x'' + 1) = 1 − ''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 1}} and {{math|1=''f''{{hsp}}(''x'' + 2<sup>''r''</sup>) = −''f''{{hsp}}(''x'')}} for {{math|0 ≤ ''x'' ≤ 2<sup>''r''</sup>}} with {{mvar|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]].
The Fabius function takes rational values at [[Dyadic rational|dyadic rationals]] that are given by the following formula:
:<math>
f\!\left(\frac s{2^n}
\right) = \sum_{p=0}^{\left\lfloor n/2\right\rfloor}\!\sum_{r=1}^s \sum_{k=0}^p\sum_{m=1}^{2^k} \frac{(-1)^{p+\nu_2(s-r)+\nu_2(m-1)}}{2^{\binom{n+1}2-\binom k2+2p(k+1)}}\cdot\frac{(2r-1)^{n-2p}}{(n-2p)!}\cdot\frac{(2m-1)^{2p+k+1}}{(2 p+k+1)!}\left(\prod_{a=1}^k\frac1{4^a-1}\!\right)\!\left(\prod_{a=1}^{p-k}\frac1{4^a-1}\!\right)\!,</math>
where <math>\nu_2(n)</math> is the sum of digits of ''n'' in [[base-2]].
==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 | issue=2 | pages=173–174 | doi=10.1007/bf00536652}}
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