Two variables — partials can exist without continuity

Suppose $f$ on $\mathbb{R}^{2}$ is given by

$$f(x,y)=\begin{cases}\dfrac{xy}{x^{2}+y^{2}},&(x,y)\neq(0,0),\\ 0,&(x,y)=(0,0).\end{cases}$$

$S_1$: $f$ is continuous at $(0,0)$. $S_2$: Both partial derivatives exist at $(0,0)$.