Definition:Continuous Real Function/Point
< Definition:Continuous Real Function(Redirected from Definition:Continuous Real Function at Point)
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Definition
Let $A \subseteq \R$ be any subset of the real numbers.
Let $f: A \to \R$ be a real function.
Let $x \in A$ be a point of $A$.
Definition 1
$f$ is continuous at $x$ if and only if the limit $\ds \lim_{y \mathop \to x} \map f y$ exists and:
- $\ds \lim_{y \mathop \to x} \map f y = \map f x$
Definition 2
$f$ is continuous at $x$ if and only if the limit $\ds \lim_{y \mathop \to x} \map f y$ exists and:
- $\ds \lim_{y \mathop \to x} \map f y = \map f {\lim_{y \mathop \to x} y}$
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.
Also defined as
Often continuity is defined only for limit points of the domain.