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> Let $f: U \to \mathbb{R}$ be a differentiable function in the open $U \subset \mathbb{R}^{n}$. If, for every $a \in \partial U$, $\lim\_{x \to a}f(x) = 0$, prove that there is at least a critical point of $f$ in $U$.
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I don't know what to do. I cannot use $\lim\_{x \to a}f(x) = f(a)$ because $a \in \partial U$. I know that $a = \lim x\_{n}$ with $x\_{n} \in U$ and I'm trying to figure out how to use it. Before, I tried to show that $U$ has a maximum or minimum point, but I couldnot develop. Can someone help me?