"We defined the notion of continuity for functions defined on a subset $E$ of a metric space $X$. However, the complement of $E$ in $X$ plays no role whatever in this definition (note that the situation was somewhat different for limits of functions). Accordingly, we lose nothing of interest by discarding the complement of the domain of $f$. This means that we may juct as well talk only about continuous mappings of one matric space into another, rather than of mappings of subsets."
I have one question:
Why $E^c$ in $X$ plays no role in definition of continuous function but it's material in definition limit of function?