# Definition:Upper Bound of Mapping/Real-Valued

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*This page is about Upper Bound of Real-Valued Function. For other uses, see Upper Bound.*

## Definition

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 $f$**.

### Upper Bound of Number

When considering the upper bound of a set of numbers, it is commonplace to ignore the set and instead refer just to the number itself.

Thus the construction:

*The set of numbers which fulfil the propositional function $P \left({n}\right)$ is bounded above with the upper bound $N$*

would be reported as:

*The number $n$ such that $P \left({n}\right)$ has the upper bound $N$*.

This construct obscures the details of what is actually being stated. Its use on $\mathsf{Pr} \infty \mathsf{fWiki}$ is considered an abuse of notation and so discouraged.

This also applies in the case where it is the upper bound of a mapping which is under discussion.

## Also see

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 7.13$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**bound**:**1.**(of a function) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**bound**:**1.**(of a function)