= 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>.
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