If $f(0) = f(n)$ then there are $n$ different $(x , y)$ such that $f(x) = f(y)$ and $x-y \in \mathbb{Z}/\{0\}$

Question

$f$ is continuous on $[0,n]$ and $f(0) = f(n)$. Prove that there are $n$ different $(x,y)$(regardless of order) such that $f(x) = f(y)$ and $x - y \in \mathbb{Z}/\{0\}$.

I have no idea how to do it.