Schwarz–Clairaut theorem

Schwarz–Clairaut theorem: Let $U\subseteq\mathbb R^n$ be an open set and let $f:U\to\mathbb R$. Fix indices $i,j\in\{1,\dots,n\}$. If the mixed second partial derivatives $\frac{\partial^2 f}{\partial x_i\partial x_j}$ and $\frac{\partial^2 f}{\partial x_j\partial x_i}$ exist on a neighborhood of $a\in U$ and are continuous at $a$, then

This justifies treating the Hessian matrix of a sufficiently smooth function as symmetric, and it is a standard hypothesis in second-order local analysis such as the second derivative tests .