Clarifying the definition of continuity at a point in Johnsonbaugh/Pfaffenberger Foundations of Mathematical Analysis In my copy of Foundations of Mathematical Analysis, in the section on continuity, I'm not understanding definition 33.1. The definition is as follows:
Let $f$ be a function from a subset X of R into R. We say that $f$ is continuous at a if either:


*

*a is an accumulation point of X and $ \lim_{x \to a} f(x) = f(a)$

*a is not an accumulation point of X.


I don't understand the second bullet point. If a is not an accumulation point of X, then can't it be that a is not in the domain of f ? If that is the case, why would we say that f is continuous at a ? I can't see how it could vacuously be the case that f is continuous there. Perhaps is the definition simply missing a stipulation that $ a \in X$ ? Thanks!

In fact, you need $a\in X$. Otherwise, there is no meaning in the continuity of $f$ outside of $X$. Consider $X=\{0\}\cup [1,2]$. Then $0\in X$ is no accumulation point of $X$. Hence, by definition $f$ is continuous at $0$.
