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Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a $C^2 $ function and $x^*$ be a point such that $\bigtriangledown^2f(x^*)$ is positive definite.Is it always true that,there exists a neighborhood around $x^*$ such that for all points $x$ in that neighborhood ,we have $\bigtriangledown ^2f(x)$ is positive definite?

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Yes: since $\nabla^2f(x^*)>0$, all the determinants of the up-left square submatrices are positive. By continuity of the partial derivatives of order $2$, the map $g_k\colon x\mapsto \det (\nabla^2f(x))_k$, where $A_k$ means the submatrix of $A=(a_{i,j})_{1\leq i,j\leq n}$, namely $A_k=(a_{i,j})_{1\leq i,j\leq k}$, is continuous.

Therefore, we can find a neighborhood $V$ of $x^*$ such that $g_k(x)>0$ for all $x$ in $V$ and each $k\in \{1,\dots,n\}$. We conclude by Sylvester's criterion.