I have one question to understanding of boundedness of periodic function. So if we have $f$ is continuous periodic function with $f(x)=f(x+p)$ for some $p>0$ and $f: \mathbb R\to \mathbb R$. Then $\exists \quad x'\in [0,p]$ such that $f(x)=f(x') \quad \forall x \in \mathbb R$. Can somebody precise explain me, why does this $x'$ exist? Because I understand it only intuitively
Thank you!