Why does the definition of the functional limit involve a limit point? This may be an odd question, and I'm not sure if these type of questions are at all appreciated in the maths community. But given the definition of the functional limit:

Let $f: A \to \mathbb{R}$, and let $c$ be a limit point of the domain $A$. We say that $\lim_{x \to c} f(x)=L$ provided that, for all $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $0<|x-c|< \delta$ (and $x \in A$) it follows that $|f(x)-L|< \epsilon$.

why does $c$ have to be a limit point? I understand that the above is just a definition, and we can define anything we want to. But, after having gone through an entire chapter of functional limits and continuity, I have not yet figured out what would go wrong if $c$ was not a limit point.

If $c$ is not a limit point of $A$, then for all small enough $\delta > 0$, the set $\{ x \in A : 0 < \lvert x-c\rvert < \delta\}$ is empty, and the condition
$$(x\in A \land 0 < \lvert x-c\rvert < \delta) \implies \lvert f(x) - L\rvert < \varepsilon$$
is vacuously satisfied for every $L$. Thus the limit would not be unique if we didn't require $c$ to be a limit point of $A$.
