Assume that $\displaystyle f_{xy}$ and $\displaystyle f_{yx}$ are continuous and that $\displaystyle f_{yxx}$ exists. Show that $\displaystyle f_{xyx}$ also exists and that $\displaystyle f_{yxx} = f_{xyx}$.
$\displaystyle f_{yxx}$ means $\displaystyle (f_{yx})_x$. Since $\displaystyle f_{xy}$ and $\displaystyle f_{yx}$ are continuous, $\displaystyle f_{yx}= f_{xy}$.