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?
Best Answer
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:
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.