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Suppose $F:\mathbb{R}^N\rightarrow \mathbb{R}^1$ is in $C^2$. Denote the Hessian of $F(x)$ at point $x$ by $H(x)$.

Let $S$ be a closed subset of $\mathbb{R}^N$ and $k_1,k_2$ be positive real values. For all $x_1,x_2 \in S$ suppose the below is true:

$$|\det(H(x_1))-\det(H(x_2))|<k_1$$

And

$$|\det(H(x_1))|>k_2,|\det(H(x_2))|>k_2$$

Denote the number of positive eigenvalues of $H(x)$ by $n(x)$. I want to prove that, for all $x_1,x_2\in S$;

$$n(x_1)=n(x_2)$$

But am unsure where to start. I believe this should be true as a proof I am working through has quoted this end result under these conditions but I am unsure how to prove it.

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