# Definition:Supremum of Mapping/Real-Valued Function

*This page is about suprema of real-valued functions. For other uses, see Definition:Supremum.*

## Contents

## Definition

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

Let $f$ be bounded above on $S$.

### Definition 1

The **supremum of $f$ on $S$** is defined by:

- $\displaystyle \sup_{x \mathop \in S} f \left({x}\right) := \sup f \left[{S}\right]$

where

### Definition 2

The **supremum of $f$ on $S$** is defined as $\displaystyle \sup_{x \mathop \in S} f \left({x}\right) := K \in \R$ such that:

- $(1): \quad \forall x \in S: f \left({x}\right) \le K$
- $(2): \quad \exists x \in S: \forall \epsilon \in \R_{>0}: f \left({x}\right) > K - \epsilon$

## Also known as

Particularly in the field of analysis, the supremum of a set $T$ is often referred to as the **least upper bound of $T$** and denoted $\map {\operatorname {lub} } T$ or $\map {\operatorname {l.u.b.} } T$.

Some sources refer to the **supremum of a set** as the **supremum on a set**.

## Also defined as

Some sources refer to the supremum as being ** the upper bound**.

Using this convention, any element greater than this is not considered to be an upper bound.

## Linguistic Note

The plural of **supremum** is **suprema**, although the (incorrect) form **supremums** can occasionally be found if you look hard enough.

## Also see

- Continuum Property, which guarantees that this
**supremum**always exists.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**bound**:**1.**(of a function)