The Fabius function is defined on the unit interval [0, 1] to satisfy the functional differential equation d/dx f(x) = 2f(2x) on [0, 1/2], and the reflection formula f(1-x) = 1-f(x), with initial condition f(0) = 0. It can be extended to the entire real line by continuing to satisfy d/dx f(x) = 2f(2x) for all x > 0, and outputting 0 for x < 0 (which trivially also satisfies the differential equation). The Fabius function is smooth (infinitely differentiable) but nowhere analytic (its Taylor series does not converge to itself on any neighborhood). At the endpoints x = 0 and x = 1 all of its derivatives are zero, yet it is not constant on any interval. At dyadic rationals it takes on rational values and its Taylor series becomes a finite-degree polynomial. Both the Fabius function and its extended version are implemented here, as f_abius(x) and f_abiusExt(x) respectively, as well as their nth derivatives f_abiusPrime(x,n) and f_abiusExtPrime(x,n). https://en.wikipedia.org/wiki/Fabius_function