# Definition:Continuous Real Function/Everywhere

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## Definition

Let $f: \R \to \R$ be a real function.

Then $f$ is **everywhere continuous** if and only if $f$ is continuous at every point in $\R$.

## Also known as

Some sources take the **everywhere** nature of such a real function as understood, and merely refer to it as a **continuous (real) function**.

## Sources

- 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 11$. Continuity on the Euclidean line - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.4$: Continuity: Definition $1.4.4$