Question 41
Let $f: \mathbb{R} \to \mathbb{R}$ be a function satisfying $$|f(x+y) - f(x) - f(y)| \lt 1 \text{ for all } x, y \in \mathbb{R}.$$ Prove that $\left| f\left(\frac{x}{2008}\right) - \frac{f(x)}{2008} \right| \lt 1$ for all $x \in \mathbb{R}$.