Equality of mixed partial derivatives under continuity hypotheses

Suppose \(f:\mathbb R^2\to\mathbb R\) is continuous, the first partial derivatives \[ \frac{\partial f}{\partial x},\qquad \frac{\partial f}{\partial y} \] exist everywhere and are continuous everywhere, and the mixed partials \[ \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right), \qquad \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) \] also exist everywhere and are continuous everywhere. Prove that \[ \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right) = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right) \] at every point of \(\mathbb R^2\).