Definition:Continuous Mapping (Topology)/Everywhere/Pointwise

Definition
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.

The mapping $f$ is continuous everywhere (or simply continuous) $f$ is continuous at every point $x \in S_1$.

Also see

 * Equivalence of Definitions of Everywhere Continuous Mapping between Topological Spaces