# Definition:Bounded Mapping/Real-Valued/Definition 3

## Definition

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

$f$ is bounded on $S$ if and only if:

$\exists a, b \in \R_{\ge 0}: \forall x \in S: \map f x \in \closedint a b$

where $\closedint a b$ denotes the (closed) real interval from $a$ to $b$.