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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?

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  • $\begingroup$ The way I see it, $xy=0$ means that either $x=0$ of $y=0$. Thus the graph of this expression is all the points of the two axes $\{(x,y)|x=0\text{ or }y=0\}$. The only thing special about $(0,0)$ is that the graph near this point looks like a cross, but near any other point it looks like a line segment $\endgroup$ Commented Feb 27, 2020 at 18:53

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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 $x$.

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  • $\begingroup$ 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 could Desmos be saying there is? $\endgroup$ Commented Feb 27, 2020 at 20:07
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    $\begingroup$ @SeanO'Gary Whatever Desmos is, evidently it doesn't understand something here. As a general rule I tend to suspect that you shouldn't try to learn math from software... $\endgroup$ Commented Feb 27, 2020 at 20:09
  • $\begingroup$ That makes sense. Need to be more careful in the future. $\endgroup$ Commented Feb 27, 2020 at 20:11
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$xy=0$ at the $x$ and $y$ axes. That is

$$\\{ (x, y) | x = 0 {\rm \ or\ } y = 0 \\}.$$

The point $(0, 0)$ is valid. There's no discontinuity.

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  • $\begingroup$ 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 could Desmos be saying there is? $\endgroup$ Commented Feb 27, 2020 at 20:09
  • $\begingroup$ Also, if I graph y(x-d)=0, this moves the apparent discontinuity to d. $\endgroup$ Commented Feb 27, 2020 at 20:10

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