Definition:Pointwise Inequality of Extended Real-Valued Functions

Definition
Let $S$ be a set, and let $f,g: S \to \overline{\R}$ be extended real-valued functions.

Then pointwise inequality of $f$ and $g$, denoted $f \le g$, is defined to hold iff:


 * $\forall s \in S: f \left({s}\right) \le g \left({s}\right)$

where $\le$ denotes the usual ordering on the extended real numbers $\overline{\R}$.

Thence pointwise inequality of extended real-valued functions is an instance of an induced relation on mappings.

Also see

 * Pointwise Inequality of Real-Valued Functions, a similar concept for real-valued functions
 * Pointwise Inequality, an abstraction replacing $\overline{\R}$ by an arbitrary ordered set