Clairaut's theorem on equality of mixed partials

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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 \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.