RO-NMO 2007 Problem 114

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$ . Let's call this property $(P)$ .

a) Show that if $f$ is a function with property $(P)$ , then there exists a real $x$ such that $f'(x)=0$ .

b) Give an example of a non-constant function $f$ with property $(P)$ .

c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.