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If I have a function with a domain that has "gaps" in it, can it still be continuous?

For example, consider the piecewise function

$$f(x)=\begin{cases}x & x\in[-1,0]\\ x-1 & x\in(1,2] \\ \end{cases}$$

With graph:

Graph of function in question

Is this continuous? From the epsilon-delta definition I would think it is, as the function is not defined in the regions where it "jumps" - but this goes against the naive high school era idea of continuity I had, involving moving a pencil along the function without lifting it off the paper.

hegash
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    Yes, it's continuous. Another surprising example is the function $x\to\dfrac1x$ defined for $x\in\Bbb R^*$, which is also continuous. Try by any means to forget the naive high school definition: it's just plain wrong. – Jean-Claude Arbaut Mar 20 '22 at 17:56
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    You maybe mean a "disconnected" domain, where (dis)connectivity has a certain topological meaning – FShrike Mar 20 '22 at 17:59
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    I have closed this question as the answers at https://math.stackexchange.com/questions/2829882/can-a-continuous-function-have-discontinuities are excellent and do in fact answer my question. It seems a shame though that the question there was phrased so poorly and is really quite confusing... – hegash Mar 20 '22 at 18:06

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