# Definition:Infimum of Mapping/Real-Valued Function

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

## Definition

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

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

### Definition 1

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

- $\ds \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$

where

### Definition 2

The **infimum of $f$ on $S$** is defined as $\ds \inf_{x \mathop \in S} \map f x := k \in \R$ such that:

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

## Also known as

Particularly in the field of analysis, the infimum of a set $T$ is often referred to as the **greatest lower bound of $T$** and denoted $\map {\mathrm {glb} } T$ or $\map {\mathrm {g.l.b.} } T$.

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

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

Some older sources, applying the concept to a (strictly) decreasing real sequence, refer to an **infimum** as a **lower limit**.

## Also defined as

Some sources refer to the infimum as being ** the lower bound**.

Using this convention, any element less than this is not considered to be a lower bound.

## Linguistic Note

The plural of **infimum** is **infima**, although the (incorrect) form **infimums** can occasionally be found if you look hard enough.

## Also see

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

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**bound**:**1.**(of a function) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**bound**:**1.**(of a function)