# Definition:Continuous Mapping (Topology)/Everywhere/Pointwise

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## 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**) if and only if $f$ is continuous at every point $x \in S_1$.

## Also known as

If it is necessary to distinguish between multiple topologies on the same set, then the terminology **$\tuple {\tau_1, \tau_2}$-continuous** can be used to define a **continuous mapping**.

## Also see

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Definition $3.1.3$ - 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction: Functions