Double limits — the standard axes counterexample

Consider the function $f:\mathbb{R}^2\to\mathbb{R}$ defined by

$$f(x,y)=\begin{cases}x^2+y^2 & \text{if } x=0 \text{ or } y=0\\ 1 & \text{otherwise}\end{cases}$$

(the paper prints the boundary case as "$x=0,\,y=0$", which is the usual shorthand for "on the two axes"; read literally as a single point it makes S2 and S3 vacuous restatements of continuity at one point and the question loses its point). Which of the following statements is/are true? $S_1$: $f$ is discontinuous at $(0,0)$. $S_2$: $\lim_{y\to0}\lim_{x\to0}f(x,y)=\lim_{x\to0}\lim_{y\to0}f(x,y)$. $S_3$: $\lim_{x\to0}f(x,y)$ is continuous at $y=0$.