does the definition of continuity require that the domain is the reals?

Question

When we are talking about continuity at $c$. We say for a given epsilon, there is a distance delta such that for all $x$ within this distance of $c$, $|f(x)-f(c)|<\epsilon$.

What if there are some points on the domain within this distance of $c$ which do not have an output, but for all other points the definition holds? What if there are no points within this distance of $c$, as in the function is an isolated point within the delta for a given epsilon? Is it still continuous?

Answer 1

No. Functions are continuous on their domains. There simply are no points outside of their domain, as far as the function is concerned. In particular, \begin{align*}F:&\mathbb{R}\setminus\{0\}\to\mathbb{R}\\&x\mapsto\frac{1}{x}\end{align*} is continuous on all of its domain.

Answer 2

Continuity can be defined in terms of the open sets (preimage of open set is open) for any function $f:X\to Y$, where $X$ and $Y$ are any topological spaces.

If $X$ and $Y$ are metric spaces, the definitions coincide.