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

$$f$$ 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.

• Can I ask the source of this problem? If you can provide a reputable source, I would be happy to put a bounty on this to encourage solutions. May 13, 2019 at 21:22