IMC 2010 · Problem 5

Suppose that for a function $f : \mathbb{R} \to \mathbb{R}$ and real numbers $a < b$ one has $f(x) = 0$ for all $x \in (a,b)$. Prove that $f(x) = 0$ for all $x \in \mathbb{R}$ if

$$\sum_{k=0}^{p-1} f\left( y + \frac{k}{p} \right) = 0$$

for every prime number $p$ and every real number $y$.