Continuity is equivalently preservation of closure inclusion, closed-set preimages, or neighborhoods.Suggest a slogan
Let \(f\colon X\to Y\) be a map of topological spaces. The following are equivalent:
\(f\) is continuous;
\(f(\cl_X(A)) \subseteq \cl_Y(f(A))\) for every \(A\subseteq X\);
\(\inverseof{f}(B)\) is closed in \(X\) for every closed \(B\subseteq Y\);
for each \(x\in X\) and each neighborhood \(V\) of \(f(x)\), there is a neighborhood \(U\) of \(x\) such that \(f(U) \subseteq V\)
[@Mun00].