Can we talk about the continuity/discontinuity of a function at a point which is not in its domain? Let us say that I have a function $ f(x)=\tan(x)$ we say that this function is continuous in its domain.
If I have a simple function like $$ f(x)=\frac{1}{(x-1)(x-2)} $$
Can we really talk about its continuity/discontinuity at $x=1$ or at $x=2$.
From what I know we can't since it is not in its domain.
But doesn't it make every function of the form $$ f(x)=\frac{1}{g(x)}$$ continuous in its domain. Where $g(x)$ is any polynomial and $g(x)=0$ at n points (lets say).

From p. 113 of Calculus by Spivak:

The function $f(x) = \sin 1/x$ is not continuous at $0$, because it is
  not even defined at $0$, and the same is true of the function $g(x) = x\sin 1/x$.

