continuous map (topological)
Continuity is the promise that a map does not tear things apart: points that start out near each other end up near each other. In calculus this was the epsilon–delta dance, but distances are exactly what topology throws away. The brilliant move is to reformulate the whole idea using only open sets, so that it makes sense in any topological space.
A map f from a space X to a space Y is continuous if the preimage of every open set is open: whenever V is open in Y, the set f-inverse(V) of all points mapping into V is open in X. Equivalently, the preimage of every closed set is closed. Note the direction — it is preimages of open sets, not images of open sets, that must be open; continuous maps need not send open sets to open sets.
This single clause recovers the familiar epsilon–delta definition on metric spaces and obeys the same comfortable rules: identity maps are continuous, compositions of continuous maps are continuous, and constant maps are continuous. It is the right notion precisely because it is built from the only structure a topological space has, namely its open sets.
The map f(x) = x^2 from the real line to itself is continuous: the preimage of an open interval (1, 4) is (-2, -1) ∪ (1, 2), which is open. But f does NOT send the open set (-1, 1) to an open set — its image is [0, 1), which is not open. Continuity controls preimages, not images.
Preimages of open sets stay open under x^2, yet an open set's image need not be open.
Continuity can be local: f is continuous if and only if it is continuous at every point, where continuity at p means every neighborhood of f(p) has a neighborhood of p mapping into it. To test continuity it suffices to check preimages of basis (or even subbasis) elements, not all open sets.