# Definition:Bounded Above Mapping/Unbounded

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< Definition:Bounded Above Mapping(Redirected from Definition:Increase Without Bound)

*This page is about mappings which are unbounded above. For other uses, see Definition:Unbounded Above.*

## Definition

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

Then $f$ is **unbounded above on $S$** iff it is not bounded above on $S$:

- $\neg \exists H \in T: \forall x \in S: f \left({x}\right) \preceq H$