Definition:Supremum of Mapping/Real-Valued Function

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

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

Also known 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.


 * Equivalence of Definitions of Supremum of Real-Valued Function


 * Definition:Infimum of Real-Valued Function


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