Clairaut's theorem on equality of mixed partials

Statement

Suppose $f$ is a real-valued function of two variables $x,y$ and $f(x,y)$ is defined on an open subset $U$ of $\mathbb {R} ^{2}$. Suppose further that both the second-order mixed partial derivatives $f_{xy}(x,y)$ and $f_{yx}$ exist and are continuous on $U$. Then, we have:

$\!f_{xy}=f_{yx}$

on all of $U$.