Definition:Infimum of Mapping

Definition
Let $f$ be a mapping defined on a subset of the real numbers $S \subseteq \R$.

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

It follows from the Continuum Property that the image of $f$ has an infimum on $S$.

Thus:
 * $\displaystyle \inf_{x \mathop \in S} f \left({x}\right) = \inf f \left({S}\right)$

Also see

 * Definition:Supremum of Mapping

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