Fabius function
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 − x ) = 1 − f ( x ) {\displaystyle f(1-x)=1-f(x)} for 0 ≤ x ≤ 1 {\displaystyle 0\leq x\leq 1} , and the functional differential equation f ′ ( x ) = 2 f ( 2 x ) {\displaystyle f'(x)=2f(2x)} for 0 ≤ x ≤ 1 / 2 {\displaystyle 0\leq x\leq 1/2} .