C^2 implies equal mixed partials

Let $U\subseteq\mathbb{R}^n$ be open and let $f:U\to\mathbb{R}$ be of class $C^2$ .

Corollary: For all $a\in U$ and all indices $i\neq j$, $\frac{\partial^2 f}{\partial x_i\partial x_j}(a)=\frac{\partial^2 f}{\partial x_j\partial x_i}(a).$

Connection to parent theorem: Apply Schwarz's theorem at each point $a\in U$. The $C^2$ hypothesis guarantees the mixed partial derivatives exist in a neighborhood and are continuous , which is exactly the hypothesis of Schwarz’s theorem.