Continuity:
A function $f : A → \mathbb{R}$ is continuous at a point $c ∈ A$ if, for all $\epsilon > 0$, there exists a $δ > 0$ such that whenever $|x − c| < δ$ (and $x ∈ A$) it follows that $|f(x) − f(c)| < \epsilon$
Functional Limits:
Let $f : A → \mathbb{R}$, and let $c$ be a limit point of the domain $A$. We say that $\lim_{x→c} f(x) = L$ provided that, for all $\epsilon > 0$, there exists a $δ > 0$ such that whenever $0 < |x − c| < δ$ (and $x ∈ A$) it follows that $|f(x) − L| < \epsilon$.
I see two differences in these definitions - firstly, for continuity, $x=c$ is a possibility and secondly, that $c \in A$, the limit point $c$ must be contained in the set for continuity.
My question is, when do we need to use this, and what are the implications of these results?
For the existence of the limit $\lim_{x\to c}f(x)$ the function $f$ needs not be defined at $x=c$, or if it is defined, the value $f(c)$ may have no relation to the value of the limit.
For $f$ to be continuous at $c$, $f$ must be defined at $x=c$, the limit $\lim_{x\to c}f(x)$ must exist, and its value must be equal to $f(c)$.