Definition:Continuous Real Function at Point/Definition 2

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 $A$.

$f$ is continuous at $x$ 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 see

 * Equivalence of Definitions of Continuous Real Function at Point