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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?


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What do you claim is the mistake exactly? He didn't state the function is continuous there. –  Git Gud Feb 2 at 13:59
@GitGud he said the function is discontinuous at the point $x=2$, which it's not true, since the function is not defined at this point. –  user42912 Feb 2 at 14:02
You're right, it would be more proper to talk about a removable singularity. –  Hans Lundmark Feb 2 at 14:04
In order for $f$ to be continuous at $x=a$, the function must in particular be defined at $a$. So, by definition, if $a$ is not in the domain, $f$ cannot be continuous at $a$; thus, such a point is logically a discontinuity. I agree with you that the correct language use is to say that $f$ is continuous if it is continuous at every point of its domain. So these seem contradictory, but they're not. –  Ted Shifrin Feb 2 at 14:04
Because in one case you're asking a specific question: Is $f$ continuous at $x=a$? No, because $a$ is not in the domain. Thus, by definition, it is not continuous at $x=a$. The function is continuous (because it's continuous at every point of its domain), but it fails to be continuous at points not in its domain. Semantic silliness, I admit, but still correct. For a student first learning calculus, it is important to understand the definition: $f$ is continuous at $x=a$ if (i) it's defined at $a$, (ii) $\lim_{x\to a} f(x)=f(a)$. Discontinuity means one of these is violated. –  Ted Shifrin Feb 2 at 14:18
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3 Answers

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.

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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 $\{2\}$.

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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.

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*same jargon in most analysis books –  Git Gud Feb 2 at 14:10
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