In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by .
This function satisfies the initial condition , the symmetry condition for , and the functional differential equation
for . It follows that is monotone increasing for , with and and and . All derivatives are zero at 0, i.e. , and are also all zero at all positive integers.
It was also written down as the Fourier transform of
by .
The Fabius function is defined on the unit interval, and is given by the cumulative distribution function of
where the are independent uniformly distributed random variables on the unit interval. That distribution has an expectation of and a variance of .
There is a unique extension of to the real numbers that satisfies the same differential equation for all x. This extension can be defined by for , for , and for with a positive integer. The sequence of intervals within which this function is positive or negative follows the same pattern as the ThueâÂÂMorse sequence.
The Rvachëv up function is closely related to the Fabius function : It fulfills the delay differential equation
(See Delay differential equation for another example.)
The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive dyadic rational arguments. For example:
with the numerators listed in and denominators in .
for , where is Euler's constant, and is the Stieltjes constant. Equivalently,
for .