I'm trying to disprove/prove the following claim:
Let $f:[0,1] \to R$ be a continious function, such that:
$$f(0.5) < f(0) < f(1) $$
Prove that there exists a local minima or maxima to $f$.
It seems so simple to prove. Since its continuous then $f(0)$ is a maxima or otherwise there must be a point near it which is, since its point which is lower then others in the given region.
Unfortunately, I failed to figure out a formal proof.