My question is if it is possible to construct a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ such that $f_x$, $f_y$ and $f_{xy}$ exists in some point, but $f_{yx}$ does not.
I tried with two results:
If $f$ differentiable at $a$ then $f_x$, $f_y$ exists at $a$.
If $f_x$ and $f_y$ exists and are continuous at $a$, then $f$ is differentiable at $a$.
I mean, trying to prove that the last construction is impossible, if $f_{yx}$ does not exists at $a$, then by the first result $f_y$ must be not differentiable at $a$ and by the second result $f_{yy}$ or $f_{yx}$ does not exist or some of them are not differentiable at $a$. But I did't find any contradiction or suggestion for one example where the construction is possible.