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 $x$ the limit $\displaystyle \lim_{y \mathop \to x} \map f y$ exists and:
 * $\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$

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

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

Also see

 * Definition:Discontinuous Real Function at Point