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Let $f : \mathbb{R}^n \rightarrow \mathbb{R}, f \in C^2$. I have to prove that for every nondegenerate critical point of $f$, there exists a neighbourhood which does not contain any further critical points.

I know that I will have to look at the Taylor formula and I'm pretty sure that I have to use that $\det (\partial_{ij} f(x)) \neq 0$ for all $x$ which are nondegenerate, that is to say the Jacobi-matrix is invertible. This makes me feel like I should use the Inverse function theorem, but to what end?

I'd be really delighted if I could get some advice. Thanks in advance.

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You're on the right track. Apply the inverse function theorem to $({\partial f \over \partial x_1}(x),...,{\partial f \over \partial x_n}(x))$ on a neighborhood of the critical point.

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    Incidentally, another way to see this is to use the fact that around a nondegenerate critical point, a smooth function $f $ looks like $\sum x_i^2 - \sum x_j^2$ (in suitable coordinates).2011-03-27