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 $a$ :
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {x - a} < \delta \implies \size {\map f x - \map f a} < \epsilon$


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

and so on.