When is the closure characterization of continuity ever really useful? A classical exercise in elementary general topology is to show that a mapping $f\colon X\to Y$ between topological spaces is continuous if and only if for any subset $A$ of $X$ we have
$$ f(\overline{A})\subset \overline{f(A)}, $$
that is the image of the closure is contained in the closure of the image.
However, I cannot recall seeing this characterization being used explictly even a single time. Are there any prominent applications? In which kind of situation can one benefit from it?
 A: There is no real "application" as such, as far as I can remember. This characterisation is more "nice to know". It confirms the rough intuition that continuous functions do not "tear away": a continuous $f$ maps a point close to $A$ ($x \in \overline{A}$) to a point close to $f[A]$ ($f(x) \in \overline{f[A]}$) and it's nice to know that this indeed characterises continuity. Also it suggests a duality, define a property
$$\forall A \subseteq X: f[\overline{A}] \supseteq \overline{f[A]}\tag{1}$$
by reversing the inclusion. It's not too hard to show that $(1)$ is in fact just equivalent to $f$ being a closed map. No application, just a nice "symmetry".
Also, there are generalisations of topological spaces called "closure spaces", of spaces $X$ endowed with a closure operation satisfying three axioms (see Wikipedia) and we can define a notion of "closure-continuity" that is the same as the characterisation we know from topology, because it's only defined in terms of the closure operator. So it can inspire generalisations like that as well.
