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I was reading this document, and I noticed on page 3, example 5, in regards to the function $y=1/x$, it says

However, it is not a continuous function since its domain is not an interval.

Pointing out that it is continuous on its domain, but since its domain is disconnected, it must therefore be classified as discontinuous.

It was my understanding that because the function $1/x$ is continuous on every point in its domain, namely $(-\infty, 0) \cup(0, \infty)$, we can safely call it continuous.

Did I misunderstand the definition of continuity?

Alec
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    You are right. The function $1/x$ is continuous. That document is talking nonsense. – jjagmath Oct 10 '21 at 12:54
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    NO : The first line of page 3 insists that the domain must be an interval for continuity. This isn't standard but applies document-wide. It's an MIT document, likely to be in wide circulation for understanding these concepts, so I hope this particularity has been discussed before. – Sarvesh Ravichandran Iyer Oct 10 '21 at 12:57
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    The author specifies on top of page 3 that by "continuous functions" he means a function whose domain is an interval and which is continuous at all points. I don't quite agree with his choice, all the more because it seems that the only difference between that definition and the previous one of a function being continuous on an interval $[a,b]$ should be that in the latter the interval is required to be compact... –  Oct 10 '21 at 12:58
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    @jjagmath: there is no nonsense as the author states his definition of a continuous function. –  Oct 10 '21 at 13:01
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    Infact, this was discussed here before, though I'm not sure I agree with the answer : a definition is an iff by convention. – Sarvesh Ravichandran Iyer Oct 10 '21 at 13:02
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    @YvesDaoust - As a reader who became confused by it, I would call it nonsense. I mean, it's perfectly fine to establish a new definition that applies within the document only. But stating it in a way that makes it sound like "this is how we define it" is confusing. The author should at the very least say "for the scope of this document, I will consider continuity to mean..." – Alec Oct 10 '21 at 13:04
  • It is common practice to state definitions in a document, and it is implicit that these hold for the document. –  Oct 10 '21 at 15:40
  • @YvesDaoust - Ok, that's fair I guess. I don't have much experience reading this type of document, so I'll defer to your judgment on this. – Alec Oct 11 '21 at 15:11

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The answer is revealed in the comments (thanks!), but for the sake of completeness, I will summarize it in answer form.

In the document, the author, at the top of page 3, re-defines continuity:

We say a function is continuous if its domain is an interval, and it is continuous at every point of that interval.

This caused some confusion for me as I'd never heard of the constraint that the domain must be an interval. Moreover, it was phrased in a way that made it seem like this was generally known ("we say"), when in fact it was a constraint that was added within the scope of the document only ("I say").

In conclusion, water is still wet, and $1/x$ is still continuous.

Alec
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