# Definition:Upper Bound of Mapping

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*This page is about Upper Bound in the context of Mapping. For other uses, see Upper Bound.*

## Definition

Let $f: S \to T$ be a mapping whose codomain is an ordered set $\struct {T, \preceq}$.

Let $f$ be bounded above in $T$ by $H \in T$.

Then $H$ is an **upper bound of $f$**.

### Real-Valued Function

The concept is usually encountered where $\struct {T, \preceq}$ is the set of real numbers under the usual ordering $\struct {\R, \le}$:

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

Let $f$ be bounded above in $\R$ by $H \in \R$.

Then $H$ is an **upper bound of $f$**.