# 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 $A$.

Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \to x} f \left({y}\right)$ exists and:

$\displaystyle \lim_{y \to x} \ f \left({y}\right) = f \left({x}\right)$

That is:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall y \in A: \left\vert{y - x}\right\vert < \delta \implies \left\vert{f \left({y}\right) - f \left({x}\right)}\right\vert < \epsilon$

## Also defined as

Often continuity is only defined for limit points of the domain.