Why do we only compute the limit of a function at a point in a CLOSED set?

Question

In my textbook, it gives a definition of limit in a topological space. Here it is:

$X \in \mathfrak{P}(E), a \in \overline{X}, f: X \rightarrow F, l \in F $

$f$ has a limit $l$ at $a$ if and only if: $$\forall \varepsilon > 0, \exists \eta > 0, \forall x \in X, \quad d_E(x,a) \leq \eta \Rightarrow d_F(f(x), l) \leq \varepsilon $$

My question is why $a$ must be in $\overline{X}$, what happen if $a$ in $X$?

Answer 1

If $a\in X$, then $a\in\overline X$.

We define this only when $a$ is in the closure of $X$ to make sure that there are points of $E$ as close to $a$ as we want. Otherwise, every element of $F$ would be a limit of $f$ at the point $a$. But with the restriction of imposing that $a\in\overline X$, the limit (if it exists) is unique.

Answer 2

The condition that $a\in\overline{X}$ includes the case that $a\in X$, since $X\subseteq\overline{X}$.

On the other hand, if $a\not\in\overline{X}$, then there exists $\eta>0$ such that $d_E(x,a)>\eta$ for all $x\in X$, and so the definition of limits becomes vacuous ($f$ has limit $l$ at $a$ for all $l$!). So it is reasonable to only care about the case that $a\in\overline{X}$.