Q:

Graph of $xy=0$ has discontinuity at $(0,0)$ - (undefined, 0).

In finding an explicit expression for y, by diving by $x$, do you implicitly assume that $x$ is not equal to zero because that would give $\frac{0}{0}$? So, there is a removable discontinuity at $(0,0)$. However, if you consider the graph to be the set of all points that satisfy the relation $xy=0$ then $(0,0)$ should be on the graph. I just don't know how to think about this. 
The main question is, how do I think about this discontinuity and how can I learn more about interesting discontinuities in general?
I was basing this whole question on my trust in Desmos which said there was a discontinuity. I just realized other graphing utilities do not say there is a discontinuity. Why would Desmos be saying there is?

A:

There are no removable discontinuities here; for that matter there's no continuity in sight. Those notions apply to functions, and the equation $xy=0$ simply does not define $y$ as a function of $x$.

