Definition:Infimum of Mapping/Real-Valued Function

Definition
Let $f$ be a real-valued function defined on a non-empty subset of the real numbers $S \subseteq \R$.

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

Then the infimum of $f$ on $S$ is defined by:
 * $\displaystyle \inf_{x \mathop \in S} f \left({x}\right) = \inf f \left[{S}\right]$

where
 * $\inf f \left[{S}\right]$ is the infimum in $\R$ of the image of $S$ under $f$

Note that this infimum always exists by the Continuum Property.

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

Also see

 * Definition:Supremum of Real-Valued Function


 * Definition:Infimum of Mapping
 * Definition:Supremum of Mapping