Let $f: \mathbb{R}^{m} \longrightarrow \mathbb{R}$ a continuous function such that there exists directional derivative in any point of $\mathbb{R}^{m}$. If $\displaystyle \frac{\partial f}{\partial u}(u)>0$ for all $u \in S^{m-1}$ then theres exists a point $a \in \mathbb{R}^{m}$ with $\displaystyle \frac{\partial f}{\partial v}(a) = 0$ for any $v \in \mathbb{R}^{m}$.
I know that $f$ attains maximum and minimum in $S^{m-1}$, because $S^{m-1}$ is a compact set and $f$ continuous. Also, I know that if $x$ is a maximum or minimum point,$f'(x)=0$ when $f:I \longrightarrow \mathbb{R}$ with $I$ a real interval. How do I use this informations? Thanks for the any help!