# 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.

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?

• 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 Commented Feb 27, 2020 at 18:53

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 $$x$$.

• 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? Commented Feb 27, 2020 at 20:07
• @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... Commented Feb 27, 2020 at 20:09
• That makes sense. Need to be more careful in the future. Commented Feb 27, 2020 at 20:11

$$xy=0$$ 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.

• 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? Commented Feb 27, 2020 at 20:09
• Also, if I graph y(x-d)=0, this moves the apparent discontinuity to d. Commented Feb 27, 2020 at 20:10