# Definition:Upper Bound of Mapping/Real-Valued

## 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 $\map P n$ is bounded above with the upper bound $N$

would be reported as:

The number $n$ such that $\map P n$ 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.