RO-NMO 2005 Problem 114

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function.

a) Prove that $f$ is continous; b) Prove that there exists an unique function $g:[0,\infty)\to\mathbb{R}$ such that for all $x\geq 0$ we have $$f(x+g(x)) = f(g(x)) - g(x) .$$