The definition of the limit for a function $f:\Omega\rightarrow{}Y$, where the image $Y$ is a topological space and the domain $\Omega$ is a subset of some other topological space $X$, is that
$$\lim_{x\rightarrow{}x_0}f(x)=y_0$$
if for all neighborhoods $V$ of $y_0$ there exists a neighborhood $U$ of $x_0$ such that $f(U\cap\Omega-\{x_0\})\subseteq{}V$. Why do we exclude $x_0$ from $U\cap\Omega$? Are we trying to avoid something?
I can see that if there exists a neighborhood $U$ of $x_0$ such that $U\cap\Omega=\emptyset$ or $U=\{x_0\}$, then the limit as $x$ approaches $x_0$ of $f(x)$ could be any $y\in{}Y$. So under this definition the limit is trivial for points not in the closure of $\Omega$ or for isolated points. Why would we only want limit points to have non-trivial limits?
Also, considering that $Y$ is not a Hausdorff space, wouldn't $f$ be automatically continuous for isolated points? It would be continuous for every non-limit point, except that non-limit points that are not isolated points are not in the domain, which is a necessary condition for continuity at a point.
Does that also mean that if $\Omega$ has an isolated point $p$ then it can't be a Hausdorff space under the subspace topology inherited from $X$. Since every limit as $x$ approaches $p$ would have more than one value?
Best Answer
We exclude $x_0$ because the limiting behavior at $x_0$ does not depend on the definition of $f$ at $x_0$. If you included $x_0$, then functions continuous away from $x_0$ with a jump at $x_0$ would not have a limit there-but they should.
This gives the strange behavior you describe when $x_0\in \Omega$ is isolated-but if we're consistent in thinking that a limit of a function is about its behavior near a point, not at the point, then this is irrelevant.
Every function is continuous at an isolated point, regardless of Hausdorffness of the range. $\Omega$ is not a subspace of $Y$.
Your last paragraph doesn't make sense: if you're inducing a topology from $Y$, this can't be affected by its pre-existing topology, including limit points. Certainly $\Omega$ could have isolated points but have Hausdorff induced topology: consider the identity map from the discrete topology to a Hausdorff topology on the same set.