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 $S_2$.

Definition by Pointwise Continuity

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

Definition by Open Sets

The mapping $f$ is continuous on $S_1$ if and only if:

$U \in \tau_2 \implies f^{-1} \left({U}\right) \in \tau_1$

where $f^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f$.

Also known as

If it is necessary to distinguish between multiple topologies on the same set, then the terminology $\left({\tau_1, \tau_2}\right)$-continuous can be used for the above.

Also see

• Equivalence of Definitions of Everywhere Continuous Mapping between Topological Spaces

• Results about continuous mappings can be found here.