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This is the current revision of this page, as edited by ~2026-16141-58 (talk) at 22:15, 13 March 2026 (โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโขโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโฆเญฆโฆโฏโฆเญฆโฆโ โฏโโฏโโฏโโฏโ โฆเญฆโฆโฏโฆเญฆโฆโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโขโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ). The present address (URL) is a permanent link to this version.
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Graph of the Fabius function on the interval [0,1].
In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius (1966).
This function satisfies the initial condition f ( 0 ) = 0 {\displaystyle f(0)=0}, the symmetry condition f ( 1 โ x ) = 1 โ f ( x ) {\displaystyle f(1-x)=1-f(x)} for โ 0 โค x โค 1 {\displaystyle 0\leq x\leq 1}โ , and the functional differential equation
f โฒ ( x ) = 2 f ( 2 x ) {\displaystyle f'(x)=2f(2x)}
for โ 0 โค x โค 1 / 2 {\displaystyle 0\leq x\leq 1/2}โ . It follows that f ( x ) {\displaystyle f(x)} is monotone increasing for โ 0 โค x โค 1 {\displaystyle 0\leq x\leq 1}โ , with f ( 1 / 2 ) = 1 / 2 {\displaystyle f(1/2)=1/2} and f ( 1 ) = 1 {\displaystyle f(1)=1} and f โฒ ( 1 โ x ) = f โฒ ( x ) {\displaystyle f'(1-x)=f'(x)} and โ f โฒ ( x ) + f โฒ ( 1 2 โ x ) = 2 {\displaystyle f'(x)+f'({\tfrac {1}{2}}-x)=2}โ .
It was also written down as the Fourier transform of
f ^ ( z ) = โ m = 1 โ ( cos โก ฯ z 2 m ) m {\displaystyle {\hat {f}}(z)=\prod _{m=1}^{\infty }\left(\cos {\frac {\pi z}{2^{m}}}\right)^{m}}
by Bรธrge Jessen and Aurel Wintner (1935).
The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of
โ n = 1 โ 2 โ n ฮพ n , {\displaystyle \sum _{n=1}^{\infty }2^{-n}\xi _{n},}
where the ฮพn are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of 1 2 {\displaystyle {\tfrac {1}{2}}} and a variance of โ 1 36 {\displaystyle {\tfrac {1}{36}}}โ .
Extension of the function to the nonnegative real numbers.
There is a unique extension of f to the real numbers that satisfies the same differential equation for all x. This extension can be defined by f(x) = 0 for x โค 0, f(x + 1) = 1 โ f(x) for 0 โค x โค 1, and f(x + 2r) = โf(x) for 0 โค x โค 2r with r a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the ThueโMorse sequence.
The Rvachรซv up function[1] is closely related to the Fabius function f: u ( t ) = { f ( t + 1 ) , | t | < 1 0 , | t | โฅ 1 . {\displaystyle u(t)={\begin{cases}f(t+1),\quad |t|<1\0,\quad |t|\geq 1\end{cases}}.} It fulfills the delay differential equation[2] d d t u ( t ) = 2 u ( 2 t + 1 ) โ 2 u ( 2 t โ 1 ) . {\displaystyle {\frac {d}{dt}}u(t)=2u(2t+1)-2u(2t-1).} (See Delay differential equation for another example.)
โข
Values
The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:[3][4]
f ( 1 ) = 1 {\displaystyle f(1)=1}
f ( 1 2 ) = 1 2 {\displaystyle f({\tfrac {1}{2}})={\tfrac {1}{2}}}
f ( 1 4 ) = 5 72 {\displaystyle f({\tfrac {1}{4}})={\tfrac {5}{72}}}
f ( 1 8 ) = 1 288 {\displaystyle f({\tfrac {1}{8}})={\tfrac {1}{288}}}
f ( 1 16 ) = 143 2073600 {\displaystyle f({\tfrac {1}{16}})={\tfrac {143}{2073600}}}
f ( 1 32 ) = 19 33177600 {\displaystyle f({\tfrac {1}{32}})={\tfrac {19}{33177600}}}
f ( 1 64 ) = 1153 561842749440 {\displaystyle f({\tfrac {1}{64}})={\tfrac {1153}{561842749440}}}
f ( 1 128 ) = 583 179789679820800 {\displaystyle f({\tfrac {1}{128}})={\tfrac {583}{179789679820800}}}
with the numerators listed in OEIS: A272755 and denominators in OEIS: A272757.
Asymptotic
log โก f ( x ) = โ log 2 โก x 2 log โก 2 + log โก x โ
log โก ( โ log โก x ) log โก 2 โ ( 1 2 + 1 + log โก log โก 2 log โก 2 ) log โก x โ log 2 โก ( โ log โก x ) 2 log โก 2 + log โก log โก 2 โ
log โก ( โ log โก x ) log โก 2 + ( 6 ฮณ 2 + 12 ฮณ 1 โ ฯ 2 โ 6 log 2 โก log โก 2 12 log โก 2 โ 7 log โก 2 12 โ log โก ฯ 2 ) + log 2 โก ( โ log โก x ) 2 log โก 2 โ
log โก x โ log โก log โก 2 โ
log โก ( โ log โก x ) log โก 2 โ
log โก x + O ( 1 log โก x ) {\displaystyle {\begin{aligned}\log f(x)&=-{\frac {\log ^{2}x}{2\log 2}}+{\frac {\log x\cdot \log(-\log x)}{\log 2}}-\left({\frac {1}{2}}+{\frac {1+\log \log 2}{\log 2}}\right)\log x-{\frac {\log ^{2}(-\log x)}{2\log 2}}+{\frac {\log \log 2\cdot \log(-\log x)}{\log 2}}\&+\left({\frac {6\gamma ^{2}+12\gamma _{1}-\pi ^{2}-6\log ^{2}\log 2}{12\log 2}}-{\frac {7\log 2}{12}}-{\frac {\log \pi }{2}}\right)+{\frac {\log ^{2}(-\log x)}{2\log 2\cdot \log x}}-{\frac {\log \log 2\cdot \log(-\log x)}{\log 2\cdot \log x}}+O!\left({\frac {1}{\log x}}\right)\end{aligned}}}
for โ x โ 0 + {\displaystyle x\to 0^{+}}โ , where ฮณ {\displaystyle \gamma } is Euler's constant, and ฮณ 1 {\displaystyle \gamma _{1}} is the Stieltjes constant. Equivalently,
log โก f ( 2 โ n ) = โ n 2 log โก 2 2 โ n log โก n + ( 1 + log โก 2 2 ) n โ log 2 โก n 2 log โก 2 + ( 6 ฮณ 2 + 12 ฮณ 1 โ ฯ 2 12 log โก 2 โ 7 log โก 2 12 โ log โก ฯ 2 ) โ log 2 โก n 2 n log 2 โก 2 + O ( 1 n ) {\displaystyle \log f!\left(2^{-n}\right)=-{\frac {n^{2}\log 2}{2}}-n\log n+\left(1+{\frac {\log 2}{2}}\right)n-{\frac {\log ^{2}n}{2\log 2}}+\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 ^{2}n}{2n\log ^{2}2}}+O!\left({\frac {1}{n}}\right)}
for โ n โ โ {\displaystyle n\to \infty }โ .
References
"A288163 โ Oeis".
Juan Arias de Reyna (2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT].
Sloane, N. J. A. (ed.). "Sequence A272755 (Numerators of the Fabius function F(1/2^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A272757 (Denominators of the Fabius function F(1/2^n))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Fabius, J. (1966), "A probabilistic example of a nowhere analytic Cโ-function", Zeitschrift fรผr Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173โ174, doi:10.1007/bf00536652, MR 0197656, S2CID 122126180
Jessen, Bรธrge; Wintner, Aurel (1935), "Distribution functions and the Riemann zeta function", Trans. Amer. Math. Soc., 38: 48โ88, doi:10.1090/S0002-9947-1935-1501802-5, MR 1501802
Dimitrov, Youri (2006). Polynomially-divided solutions of bipartite self-differential functional equations (Thesis).
Arias de Reyna, Juan (2017). "Arithmetic of the Fabius function". arXiv:1702.06487 [math.NT].
Arias de Reyna, Juan (2017). "An infinitely differentiable function with compact support: Definition and properties". arXiv:1702.05442 [math.CA]. (an English translation of the author's paper published in Spanish in 1982)
Alkauskas, Giedrius (2001), Dirichlet series associated with ThueโMorse sequence, preprint.
Rvachev, V. L.; Rvachev, V. A. (1979), Non-classical methods of the approximation theory in boundary value problems (in Russian), Kiev: Naukova Dumka
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This page was last edited on 13 March 2026, at 22:15 (UTC).
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