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 if and only if:

$\forall s \in S: \map f s \le \map g s$

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