# Definition:Continuous Real Function at Point/Also presented as

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## Continuous Real Function at Point: Also presented as

While **continuity at a point** is compactly defined as a direct consequence of the nature of limit at that point, it is commonplace in the literature to include whatever definitions for limit in the actual **continuity** definition, for example:

- $f$ is continuous at $a$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {x - a} < \delta \implies \size {\map f x - \map f a} < \epsilon$

- $f$ is continuous at $a$ if and only if:
- $\ds \lim_{x \mathop \to a^-} \map f a$ and $\ds \lim_{x \mathop \to a^+} \map f a$ both exist and both are equal to $\map f a$

and so on.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity: Definition $1.4.3'$