# Local maxima or minima of a continuous functions

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.

• A much more general result holds. Having a continuous function on a compact domain alone grants the existence of local minima and maxima without any additional conditions. The name of this result is "extreme value theorem". – joedoe8585 Jan 23 '16 at 17:46

Since $f$ is continuous on $[0,1]$, it has at least one global minimum in this interval. Since $f(0.5)\lt f(0)$ and $f(0.5)\lt f(1)$, any such global minimum is not at an endpoint, so it must be a local minimum.