A mistake in Stewart's book I'm making a revision of calculus and I'm using Stewart's book
I think he is wrong in case $(a)$:

In case $(a)$ the function is not defined in $x=2$, then we can't say that the function is continuous or not at this point, in fact this function is continuous. Am I right?
Thanks
 A: A motive for using deleted neighborhoods $0 < |x-a| < \delta$ to define limits is to be able to separate the conditions of having a limit and having a definition for $f(x)$ at $x=a$.
So I don't have an objection to saying in case (a) that $f(x)$ is not continuous at $x=2$, because to be continuous at a point requires both a limit and a definition that exist and that these agree.
Calling a point a discontinuity where the definition does not exist, but the limit does, is convenient even if it conflicts with an expectation that continuity is defined only for points in the domain.  In other contexts mathematicians would label this a singularity (albeit a removable singularity in case (a)) without confusion.
A: Yes you are right. The domain of $f(x)$ is $\mathbb{R} - \{2\}$. So it doesn't make sense talking about continuity at the point $\{2\}$.
A: Ok, maybe it's sloppy language, but you'll find the same jargon in most books. We're not really interested in the domain of the function here as much as where the function CAN be defined so as to be continuous.
