One partial derivative is continuous (at a single point) implies differentiable? Let $f:\mathbb{R}^2\to\mathbb{R}$ and $(p,q)\in\mathbb{R}^2$ such that both $f_x$ and $f_y$ exists at $(p,q)$.
Assume that $f_x$ is continuous at $(p,q)$.
How do we prove/disprove that $f$ is differentiable at $(p,q)$?

I do note that it is similar to this question: Continuity of one partial derivative implies differentiability
However the critical difference is that for my case, I only have $f_x$ continuous at a single point $(p,q)$, not even in a neighborhood, hence I believe that the approach of Fundamental Theorem of Calculus used in the other question cannot work.
Thanks for any help!

A simpler example:  f(x,y)= 1 if xy= 0, otherwise, f(x,y)= 0.  Both partial derivatives at (0, 0) are 0 but f is not differentiable at (0, 0).
