Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$, and let $x\in U$. Then the following hold.

1. (Equality of mixed partial derivatives) For all $i,j\in\{1,\dots,n\}$,

with the second-order partial derivative notation of C^2 Real-Valued Map on an Open Subset of Euclidean Space.

2. (Symmetry of the Hessian) The Hessian matrix $D^2f(x)$ belongs to $\mathcal{S}(n)$, the set of symmetric real $n\times n$ matrices.