I am trying to understand Rudin's remark 4.12 on pages 88-89.
What Rudin says is that, though we can talk about continuity of $f$ on $E \subset X$, the complement $E^c \subset X$ plays no role, unlike in the limit definition. I'm not able to understand exactly why this is, other than that we restricted ourselves in the case of limits to letting $x \to p $ for a limit point $p$. I'm assuming the continuity definition is "vacuously true" for $x \in E^c$, but I'm not totally sure.
Could someone shed some light on this?
