Spring 2018, problem 67
Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ $$\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. $$
Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ $$\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. $$