Definition:Continuous Mapping (Topology)/Everywhere

Definition
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ 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) iff $f$ is continuous at every point $x \in S_1$.

Definition by Open Sets
Equivalently, continuity of $f$ can be defined using open sets:

That these conditions are equivalent is proven in Continuous Mapping by Open Sets.

Also see

 * Continuous Mapping by Open Sets