Partial derivatives are zero in some point, when the limit in boundary points are zero.

Question

Let $U\subset \mathbb{R}^n$ an open set and $f:U\to \mathbb{R}$ a continuous function such as his first order partial derivatives there exist in $U$. Let suppose that $$\lim_{x\to a}f(x)=0$$ for all $a\in \partial U$. Show that there is $c\in U$ such as, $$\dfrac{\partial f}{\partial x_i}(c)=0$$ for all $i\in \{1,\dots,n\}$.

Answer 1

I believe you need to include the condition that $U$ is bounded as well. Otherwise we can construct counterexamples like $U=\{(x,y)\in\mathbb R^2|x>0,y>0\}$, and $f(x,y)=xy$.

Hint: We can extend $f$ to a continuous function $\bar f:\overline U\to\mathbb R$, such that $\bar f(x)=0$ for any $x\in\partial U$. Since $\overline U$ is closed and bounded, it is compact. So we can apply extreme value theorem, there exists $c_1,c_2 \in \overline U$ such that $\bar f(c_1)\geq \bar f(x)$ and $\bar f(c_2)\leq \bar f(x)$ for any $x\in \overline U$.