Def.: let be $(A,\tau)$,$(C,\zeta)$ two topological spaces, $f \in C^E$, with $E \subseteq A$, and $x_0$ an accumulation point of $E$, a point $l \in C$ is limit of $f$ as $x$ approaches $x_0$ if $$\forall Y \in \displaystyle\mathcal{U}_{(C,\zeta)}(l) (\exists X \in \displaystyle\mathcal{U}_{(A,\tau)}(x_0)(f((X-\{x_0\} )\cap E) \subseteq Y))$$
$\displaystyle\mathcal{U}_{(A,\tau)}(x_0)$ is neighbourhood system for $x_0$
$\displaystyle\mathcal{U}_{(C,\zeta)}(l)$ is neighbourhood system for $l$
Is correct? Thanks in advance!
Best Answer
Yes, that is a correct definition of the limit of $f$ at $x_0$.
Another way to put the definition is to say that $\tilde{f}\colon E\cup \{x_0\} \to C$ defined by
$$\tilde{f}(x) = \begin{cases} f(x) &, x \in E\setminus\{x_0\} \\ l &, x = x_0\end{cases}$$
is continuous at $x_0$.