nLab Fabius function

Definition

The Fabius function is a smooth function on the unit interval that is nowhere analytic. It is defined between $0 \leq x \leq \frac{1}{2}$ by the differential equation $f'(x) = 2 f(2 x)$ with initial condition $f(0) = 0$, and by $f(1 - x) = 1 - f(x)$ for $x$ in the unit interval. The Fabius function is an example of a bump function.

In constructive mathematics

In constructive mathematics, if the analytic lesser limited principle of omniscience does not hold for the real numbers, then the Fabius function as defined above is not defined on the entire interval, since $0 \leq x \leq \frac{1}{2}$ or $\frac{1}{2} \leq x \leq 1$ is not provably logically equivalent to $0 \leq x \leq 1$ in the absence of the analytic LLPO. However, there might be a way to constructively patch the two functions together using open intervals and a third function in the same way that one might patch the function $\mathrm{exp}(\frac{1}{2} \mathrm{ln}(x))$ defined on the positive real numbers using the exponential function and logarithm to form the square root function on the non-negative real numbers. The author does not know if the resulting patched Fabius function on the entire unit interval remains smooth and nowhere analytic.

Related concepts

• smooth function

• analytic function

• bump function

References

• Jaap Fabius, A probabilistic example of a nowhere analytic $C^\infty$-function, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5 (2): 173–174, 1966 (doi:10.1007/bf00536652, MR:0197656, SCID:122126180)

• Juan Arias de Reyna, Definición y estudio de una función indefinidamente diferenciable de soporte compacto, Rev. Real Acad. Ciencias 76 (1982) 21-38. English translation: An infinitely differentiable function with compact support: Definition and properties (arXiv:1702.05442)

• Juan Arias de Reyna, Arithmetic of the Fabius function (arXiv:1702.06487)

See also:

• Wikipedia, Fabius function