Let $f \in C^2(A,\mathbb{R})$ where $A \subset \mathbb{R}^n$ is open. I have to prove that if $x_0$ is a non-degenerate critical point, than it is isolated, without using the "local inversion theorem". My idea is to suppose that $x_0$ isn't isolated and to show that this brings to a contraddiction.
So if I suppose $x_0$ isn't isolated, there $\exists$ a sequence $x_n \to x_0$ where $\nabla f(x_n) = 0$. But we know from the definition of partial derivative that $\displaystyle{\frac{\partial^2f}{\partial x_i\partial x_j}(x_0)} = \lim_{n \to \infty}\displaystyle{\frac{f_{x_j}(x_n)-f_{x_j}(x_0)}{x_{n_i}-x_{0_i}}}$. This limit is $0$ because $f_{x_j}(x_n)$ and $f_{x_j}(x_0)$ are equal to $0$, because $\nabla f(x_n) = \nabla f(x_0) = 0 \in \mathbb{R}^n$. If every second partial derivative in $x_0$ is $0$, the Hessian matrix in $x_0$ is only made of $0$, and this is a contraddiction beacuse $detH_f(x_0) \neq 0$ for hypothesis.
Is this right?
