# Definition:Continuous Real Function/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.