# Definition:Supremum of Mapping/Real-Valued Function

*This page is about Supremum in the context of Real-Valued Function. For other uses, see Supremum.*

## 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:

- $\displaystyle \sup_{x \mathop \in S} \map f x := \sup f \sqbrk S$

where

### Definition 2

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

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

## 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 $\map {\mathrm {lub} } T$ or $\map {\mathrm {l.u.b.} } T$.

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

Some sources refer to the **supremum of a set** as the **join of the set** and use the notation $\bigvee S$.

Some sources introduce the notation $\displaystyle \sup_{y \mathop \in S} y$, which may improve clarity in some circumstances.

Some older sources, applying the concept to a (strictly) increasing real sequence, refer to a **supremum** as an **upper limit**.

## 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

- 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)