Definition:Pointwise Inequality of Extended Real-Valued Functions

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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