$f(x,y) := \begin{cases} \frac{xy}{x^2+y^2}, &\mbox{if } (x,y) \neq (0,0)\\ 0, & \mbox{if } (x,y)=(0,0) \end{cases}$ not differentiable in $(0,0)^T$
If $F(0,0)=0$ and $F(x,y)= \frac{xy}{x^2+y^2}$ for $(x,y)\neq (0,0)$ then $F$ is differentiable at $(0,0)$?