Graphically, it passes the vertical line test. However, I am confused because for $x=0$, there are infinite $y$ values that satisfy $xy=0$. Can someone help me understand this?
$\endgroup$ 92 Answers
$\begingroup$It is not a function (as it was pointed out in a comment), but a relation, like a circle (a set of ordered pairs): $R=\{(x,y)|xy=0\}$.
$\endgroup$ $\begingroup$As a function of one variable it is not (in general) a function, indeed recall that by definition
$$f:A\subseteq\mathbb{R}\to \mathbb{R}\quad\forall x\in A \quad \exists! y \quad y=f(x)$$
and, as noticed by Rushabh Mehta, for $x=0$ we have that $y$ can assume any value.
It would be a function, in implicit form, if we assume for example
$$f:\mathbb{R}\setminus\{0\}\to \mathbb{R}$$
which correspons to the function $y=0$ with domain $x\neq 0$.
Refer also to the related
$\endgroup$
