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
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 ≤ 0, f (x + 1) = 1 − f (x) for 0 ≤ x ≤ 1, and f (x + 2r) = −f (x) for 0 ≤ x ≤ 2r with 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.
Values
The Fabius function is constant zero for all non-positive arguments, and assumes rational values at positive
dyadic rational arguments.
References
Fabius, J. (1966), "A probabilistic example of a nowhere analytic C∞-function", Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174,
doi:
10.1007/bf00536652,
MR0197656,
S2CID122126180
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 Thue-Morse sequence",
preprint.
Rvachev, V. L., Rvachev, V. A., "Non-classical methods of the approximation theory in boundary value problems", Naukova Dumka, Kiev (1979) (in Russian).