If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$?

Question

If $f\in C[0,2\pi]$ and $f(0)=f(2\pi)$ then $f(\theta)=f(\theta+\pi)$ for some $\theta\,\in\,(0,\pi)$?

Intuitively I think it's wrong, but I failed to come up with a single counterexample. Can anybody give a hint for proving or disproving it?

Answer 1

$$f(x)=\sin x$$

It's continuous and $f(0) = f(2 \pi)$, but there exists no $\theta \in (0, \pi)$ such that $f(\theta)=f(\theta+\pi)$.