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where the {{math|''ξ''<sub>''n''</sub>}} are [[independence (probability)|independent]] [[uniform distribution (continuous)|uniformly distributed]] [[random variable]]s on the [[unit interval]]. That distribution has an expectation of <math>\tfrac{1}{2}</math> and a variance of <math>\tfrac{1}{36}</math>.
There is a unique extension of {{mvar|f}} to the real numbers that satisfies the same differential equation for all ''x''. This extension can be defined by {{math|1=''f''{{hsp}}(''x'') = 0}} for {{math|''x'' ≤ 0}}, {{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 ''Rvachëv up function''<ref>{{cite web | url=https://oeis.org/A288163 | title=A288163 - Oeis }}</ref> is closely related: <math display="block"> u(t)=\begin{cases} F(t+1),\quad |t|<1 \\ 0, \quad |t|\geq 1 \end{cases}</math> which fulfills the [[Delay differential equation]]<ref>{{cite arXiv | eprint=1702.06487 | author1=Juan Arias de Reyna | title=Arithmetic of the Fabius function | year=2017 | class=math.NT }}</ref>
<math display="block">\frac{d}{dt}u(t)=2u(2t+1)-2u(2t-1).</math> (see [[Delay differential equation|Another example]]).
==Values==
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with the numerators listed in {{OEIS2C|A272755}} and denominators in {{OEIS2C|A272757}}.
==Asymptotic==
<math>\begin{align}\log f(x)&=-\frac{\log^2x}{2\log2}+\frac{\log x\cdot\log(-\log x)}{\log2}-\left(\frac12+\frac{1+\log\log2}{\log2}\right)\log x -\frac{\log^2(-\log x)}{2\log2}+\frac{\log\log 2\cdot\log(-\log x)}{\log2}\\&+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2-6\log^2\log2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)+\frac{\log^2(-\log x)}{2\log2\cdot\log x}-\frac{\log\log2\cdot\log(-\log x)}{\log2\cdot\log x}+O\!\left(\frac1{\log x}\right)\end{align}</math>
for <math>x\to0^+,</math> where <math>\gamma</math> is [[Euler's constant]], and <math>\gamma_1</math> is the [[Stieltjes constants|Stieltjes constant]]. Equivalently,
<math>\log f\!\left(2^{-n}\right)=-\frac{n^2\log2}2-n\log n+\left(1+\frac{\log2}2\right)n -\frac{\log^2n}{2\log2}+\left(\frac{6\gamma ^2+12\gamma_1-\pi^2}{12\log 2}-\frac{7\log 2}{12}-\frac{\log\pi}2\right)-\frac{\log^2n}{2n\log^22}+O\!\left(\frac1n\right)</math>
for <math>n\to\infty.</math>
==References==
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