$f$ is continuous between $[0,1]$, and $f(0)=f(1)$.
I want to prove that there is an $a \in [0,0.5]$ such that $f(a+0.5)=f(a)$.
ok, so Rolle's theorem can be useful here, but I can't see the connection to the derivative,
(Weierstrass, Uniform continuity?) I'll be glad to instructions.