# Definition:Continuous Real Function/Point

## 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.