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==Values==
The Fabius function is constant zero for all negative arguments, and takes rational values at non-negative [[Dyadic rational|dyadic rationals]].
:<math>
f\!\left(\frac n{2^m}\right) = \sum_{\lambda=0}^{\left\lfloor m/2\right\rfloor}\!\sum_{\mu=0}^\lambda\sum_{\nu=1}^n\sum_{\kappa=1}^{2^\mu}\frac{(-1)^{\lambda+s_2(n-\nu)+s_2(\kappa-1)}}{2^{\binom{m+1}2-\binom\mu2+2\lambda\,(\mu+1)}}\cdot\frac{(2\nu-1)^{m-2\lambda}}{(m-2\lambda)!}\cdot\frac{(2\kappa-1)^{2\lambda+\mu+1}}{(2\lambda+\mu+1)!}\!\left(\prod_{\sigma=1}^\mu\frac1{4^\sigma-1}\!\right)\!\!\left(\prod_{\sigma=1}^{\lambda-\mu}\frac1{4^\sigma-1}\!\right)\!,</math>
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