 Closure characterization of continuity

A function $f:X\to Y$ between topological spaces is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$ for every subset $A\subseteq X$, where the bars denote closure. For a continuous $f$, pull back an open neighbourhood of $f(x)$ for $x\in \overline{A}$; its preimage meets $A$, so the neighbourhood meets $f(A)$. Conversely, apply the inclusion to $A=f^{-1}(C)$ with $C$ closed. Then $f(\overline{A})\subseteq C$ implies $\overline{A}\subseteq A$, so the preimage of every closed subset is closed and $f$ is continuous.