Let $m\in\mathbb{N}$, let $U\subset\mathbb{R}^m$ be open, let $f:U\to\mathbb{R}$ be a function, let $a\in U$, and let $i,j\in\{1,\ldots,m\}$. Suppose there exists an open neighbourhood $V\subset U$ of $a$ such that the first partial derivatives $\partial_{x_i}f$ and $\partial_{x_j}f$ exist at every point of $V$, the iterated partial derivatives $\partial_{x_i}(\partial_{x_j}f)$ and $\partial_{x_j}(\partial_{x_i}f)$ exist at every point of $V$, and the functions

\begin{align*} \partial_{x_i}\partial_{x_j}f:V\to\mathbb{R} \end{align*}

and

\begin{align*} \partial_{x_j}\partial_{x_i}f:V\to\mathbb{R} \end{align*}

are continuous at $a$. Then

\begin{align*} \partial_{x_i}\partial_{x_j}f(a)=\partial_{x_j}\partial_{x_i}f(a). \end{align*}