Let $\displaystyle E\subset$$\displaystyle \mathbb{R}^n$ and $\displaystyle f: E\rightarrow\mathbb{R}$ be a continuous function. Prove that if $\displaystyle a$is a local maximum point for $\displaystyle f$, then either $\displaystyle f$ is differentiable at $\displaystyle x = a$ and $\displaystyle Df(a) = 0$ or $\displaystyle f$ is not differentiable at $\displaystyle x = a$. Deduce that if $\displaystyle f$ is differentiable on $\displaystyle E^o$, then a global maximum point of f is either a critical point of f or an element of $\displaystyle \partial E$.

It's a little bit about optimization but stil analysis. Well I have no idea about this question and I think I need a proof. Thank you!