MathNet 2007-part-01 — 187 problems (EN–VI) Question 148

Let $f(x)$ be a function satisfying $f(x + 1) - f(x) = 2x + 1$ ($x \in \mathbb{R}$), and $|f(x)| \le 1$ when $x \in [0, 1]$. Prove:
$$
|f(x)| \le 2 + x^2 \quad (x \in \mathbb{R}).
$$