Where is $f(x,y)=\frac{xy}{x^2-y^2}$ (0 at singularities) differentiable? How about $g(x,y)=\frac{x|y|}{\sqrt{x^2+y_2}}$ (0 at singularity)?
I am not just looking for answers here, but an algorithm that always works. This is kind of a remedial question for me. I can compute the partials, but I don't recall if it suffices to check continuity along the axes, or whether a more general argument is needed. And beyond that, whether there is a better approach altogether, such as the abstract definition of derivative or looking at the directional derivatives.