Continuity via open sets

Continuity via open sets: Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and $f:X\to Y$. Then $f$ is continuous (at every point) if and only if for every open set $V\subseteq Y$, the preimage $f^{-1}(V)=\{x\in X: f(x)\in V\}$ is open in $X$.

This formulation is fundamental in topology and makes continuity compatible with compositions and other structural operations.

Proof sketch (optional): If $f$ is continuous and $x\in f^{-1}(V)$, then $f(x)\in V$ and some ball around $f(x)$ lies in $V$; continuity gives a ball around $x$ mapped into that ball, hence into $V$. Conversely, take $V$ to be an open ball around $f(x)$ to recover the $\varepsilon$–$\delta$ definition.