CN-SMO 2007 Problem 5

Let $f(x)$ be a function satisfying $f(x+1)-f(x)=2x+1 (x \in \mathbb{R})$ .In addition, $|f(x)|\le 1$ holds for $x\in [0,1]$ . Prove that $|f(x)|\le 2+x^2$ holds for $x \in \mathbb{R}$ .