Recently I proved this fact:
If in some neighborhood of the point $(x, y)$ there exists $f_x$, $f_y$, $f_{xy}$, and if $f_{xy}$ is continuous in $(x, y)$, then there exist $f_{yx}$ in $(x, y)$ and $f_{xy} = f_{yx}$.
I'm trying to find an example when there exist $f_x$, $f_y$, $f_{xy}$ in some neighborhood of $(x, y)$, and $f_{xy}$ is continuous for $x$ with fixed $y$ in $(x, y)$ but $f_{xy} \neq f_{yx}$.
