Definition:Bounded Mapping/Real-Valued

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

Then $f$ is bounded iff there exists a non-negative real number $K$ such that:
 * $\forall x \in S: \left\vert{f \left({x}\right)}\right\vert \le K$

$K$ is called a bound for $f$ on $S$.

Also see

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