Definition:Continuous Real Function/Everywhere

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): $1$: Review of some real analysis: $\S 1.4$: Continuity: Definition $1.4.4$
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.2$: Continuous and linear maps. Continuous maps