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.

To see where things go wrong take $A=\mathbb{N}$ and some function
$f:A\to\mathbb{R}$ and say you want to prove that 
$$
\lim_{x\to c}f(x)=L
$$
where $c\in A$ is a natural number.
Let $\epsilon>0$ and choose $\delta(\epsilon)=\delta=\frac{1}{2}$
Indeed - For every $x\in A$ s.t $0<|x-c|<\frac{1}{2}$ we have it
that $|f(x)-L|<\epsilon$
This is true vacuously because there are no such $x\in A$ that satisfy
$0<|x-c|<\frac{1}{2}$ so we get 
$$
\lim_{x\to c}f(x)=L
$$
regardless of what $L$ is, which is unreasonable 
