Continuity in topological spaces in terms of closures. Given spaces $(X, T_x)$ and $(Y, T_y)$ as well as a map $f: X \rightarrow Y $.
We say $f$ is continuous in $x \in X$, if:
$$\forall A \subset X, x \in cl_X(A) \Rightarrow f(x) \in cl_Y(f[A])$$
Since the following holds for any map:
$$\forall A \subset X,x \in A \Rightarrow f(x) \in f[A] \subset cl_Y(f[A])$$
It seems then that the key aspect in the definition is what the map does to points $x \in fr_X(A)$.
Do I understand it right that when looking at whether $f$ is continuous or not, it is sufficient to look at what happens on the $fr_X(A)$ (for any $A$ respectively) in order to see whether those points are mapped in such a way that the result is included in $cl_Y(f[A])$?
 A: Yes, it is indeed an equivalence that $f$ is continuous iff $f(\bar{A}) \subseteq \overline{f(A)}$ for any $A \subseteq X$ (by $\bar{A}$ I denote $cl_X(A)$).
So indeed, you do need to check that all the points of $A$ are mapped to the closure of $f(A)$, but that's not enough. You also need to make sure that this holds for all the points of the closure of $A$ (which are not necessarily in $A$).
A: Indeed, if you'd want to check continuity with the criterion 
$$f: X \to Y \text{ is continuous } \iff \forall A \subseteq X: f[\overline{A}] \subseteq \overline{f[A]}$$
you'd "only" have to focus your attention (for any given $A$) on the points of $\overline{A}\setminus A$ (a set sometimes called the frontier of $A$, which is in general different from the boundary of $A$ (which equals $\overline{A} \cap \overline{A^\complement}= \overline{A}\setminus \operatorname{int}(A)$)), because those in $A$ already trivally map to $f[A] \subseteq \overline{f[A]}$; it's about the points "close to" $A$, but not in $A$ already, really: we want to see that those points (close to $A$) stay close to $f[A]$ after applying $f$, so "no tearing", as it were. 
I know of no practical case where continuity is actually checked in this way; the left to right implication is used quite often, e.g. to see that a dense set of $X$ maps to a dense set of $f[X]$, etc.
