Is it possible to have a function $f:G\subseteq \mathbb R^2\to\mathbb R$, differentiable in a point $(x,y)\in G$, such that one of its partials is not continuous at $(x,y)$?
I think that it is not possible to find such a function $f$. Because if $f$ is differentiable at some point $(x,y)$, then it is continuous at that point.
Also $f'$ must be equal to ($\partial f_1,\partial f_2$).
Is my argumentation correct?