Prove the existence or not existence of continuity, differentiability and the continuity of partial derivatives of the following function two variables at $(0,0)$
$$f(x,y) = \begin{cases} (x-y)^{2}sin \frac{1}{x^{2}-y}, & \mbox{if } (x,y) \neq (0,0) , \\ 0, & \mbox{if } (x,y) =(0,0)\end{cases}$$
At first I wanted to prove this function was not continuous at $(0,0)$ therefore I would prove this function is not differentiable at $(0,0)$, as well. I tried to compute the limit of the function as we aproach to $(0,0)$ with two different trajectories,lets say $y=x$ and $y=2x$ but this doesnt seems to work. So maybe this function is continuous and differentiable. But I cant see the $\epsilon- \delta$ trick here. In order to prove differentiability I need to compute partial derivatives and show this partial derivatives are continuous isnt? But its seems harder to prove continuity in the partia derivatives since I cannot even compute continuity over the original function. :(
As $\frac{\partial f}{ \partial x}(0,0)=lim_{h \to 0} \frac{f(h,0)-f(0,0)}{h}= lim_{h \to 0} \frac{f(h,0)}{h}=lim_{h \to 0} \frac{(h)^{2}sin \frac{1}{h^{2}}}{h} $
but I dont know how to reduce this into something that gives me an insight about the differentiability or continuity of this partial derivative. In the same way I got
$\frac{\partial f}{ \partial y}(0,0)=\frac{h^{2}sin \frac{1}{-h^{2}}}{h}.$