Let $f:[0,1]\to\mathbb{R}$ continuous such that $f(0)=f(1)$. Is it true that $\forall\alpha\in(0,1)\exists c\in[0,1-\alpha]|f(c)=f(c+\alpha)$?
At first I tried to find a counterexample but my intuition says it's true. Then I've got the idea of applying Bolzano's Theorem to $g(x)=f(x)-f(x+\alpha)$ defined on $[0,1-\alpha]$ but I didn't get anything. What can I do?