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.

Also defined as
Some sources refer to the infimum as being the lower bound. Using this convention, any element greater 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

 * Definition:Supremum of Real-Valued Function


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