Existence of partials for all vectors $p$ at origin implies continuity

Question

If the partial derivative $\frac{\partial f}{\partial p}(0)=0$ for all vectors $p$ is $f$ necessarily continuous at the origin. I would say no, but it is difficult to come up with an example. Any suggestions?

Answer 1

No, indeed. Take, for instance,$$f(x,y)=\begin{cases}0&\text{ if }xy=0\\1&\text{ otherwise.}\end{cases}$$I suppose that you can check that $\frac{\partial f}{\partial x}(0,0)=\frac{\partial f}{\partial y}(0,0)=0$ and that $f$ is discontinuous at $(0,0)$.