I have to solve the following exercise:
The function $f : \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with $P = 2\pi$ so that $f(x) = f(x + 2\pi)$ is true for all $x \in \mathbb{R}$. Show that there is a $\xi \in \mathbb{R}$ with $f(\xi) = f(\xi + \pi)$.
Either I don't understand the exercise or the exercise doesn't make sense. Sure, for $f(x) = sin(x)$ for example there is a $\xi \in \mathbb{R}$. But not all periodic functions have a $\xi$.
For example the function $f(x) = mod(x,2\pi)$ don't have a $\xi$.
Am I right or did I understand something wrong?