Definition:Continuous Real Function at Point/Also presented as

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 $1.4.3'$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): above