Continuity Defined by Closure

Theorem
Let $T_1 = \left({X_1, \vartheta_1}\right)$ and $T_2 = \left({X_2, \vartheta_2}\right)$ be topological spaces.

Let $f: T_1 \to T_2$ be a mapping.

Then $f$ is continuous iff:
 * $\forall H \subseteq X_1: f \left({H^-}\right) \subseteq \left({f \left({H}\right)}\right)^-$

where $H^-$ denotes the closure of $H$ in $T_1$.

That is, iff the image of the closure is a subset of the closure of the image.