I came across the problem which is as follows:
Define $f:\mathbb R^{2}\rightarrow \mathbb R$ by $\begin{align}f(x,y)&=\begin{cases}1\,\,\,\, \text{if xy=0}\\ 2 \,\,\, \,\text{otherwise}\end{cases} \end{align}$ Now, if $S=\lbrace(x,y):f\mbox { is continuous at} (x,y)\rbrace$, then which of the following is correct?
(a) $S$ is open,
(b) $S$ is connected,
(c) $S$=$\phi$,
(d) $S$ is closed.
My attempts: Here,we see that points of discontinuity lie on the $x$ and $y$-axis where one of x and y is zero.So,I can come to conclusion that the set of points of $S$ (where $f$ is supposed to be continuous)lie in the co-ordinate planes minus the coordinate axes.So, I think the set S is connected.Am I going in the right direction? I need a bit of explanation here. Please help.Thank you in advance for your time.