Definition:Bounded Mapping/Real-Valued

Definition
Let $f: S \to \R$ be a real-valued function.

Then $f$ is bounded if there is a number $K \ge 0$ such that:
 * $\forall x \in S: \left|{f \left({x}\right)}\right| \le K$

Also see

 * Bounded Set of Real Numbers‎ for a demonstration that this definition is compatible with boundedness on an ordered set.