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The following is the definition for continuity at $x_0$
$$\forall \quad\varepsilon > 0\ \quad \exists \delta > 0\ \quad \text{s.t. } \quad |x - x_0| < \delta \implies |f(x) - f(x_0)| < \varepsilon $$
But suppose that that $f(x_0)$ is undefined, i.e. there is a "hole" at $f(x_0)$, the definition still holds for continuity since it does not care about what is happening at $f(x_0)$, only around it. Isn't this a problem?
Your definition is missing something. A function $f: X\to Y$ is continuous at $x_0\in X$ if $\dots$. That $x_0$ is in the domain of the function (i.e. that the function is defined at $x_0$) is part of the definition.
See for example the Wikipedia article on continuous functions.
What do you think of this definition?
$\forall \ \varepsilon > 0\ \ \exists \delta > 0\ \text{s.t. } \ |x_1 - x_0| < \delta \ \land\ |x_2 - x_0| < \delta \implies |f(x_1) - f(x_2)| < \varepsilon $
Yes, of course. The only thing that matters for continuity are those points at which the function is defined. What occurs outside the domain is irrelevant. If $f(x_0)$ is not defined, then we do not care about it.
