Partial derivatives fail to exist for this classic non-differentiable example

Let $f$ be a function from $\mathbb{R}^2$ to $\mathbb{R}$ given by $f(x,y)=x$ if $x=y$, $f(x,y)=1$ if $x\neq y$. Then, which of the following statements is true? $S_1$: Both partial derivatives of $f$ at $(0,0)$ exist and are equal. $S_2$: $f$ is differentiable at $(0,0)$.