Definition:Absolute Value of Mapping/Extended Real-Valued Function

Definition
Let $S$ be a set, and let $f: S \to \overline \R$ be an extended real-valued function.

Then the absolute value of $f$, denoted $\size f: S \to \overline \R$, is defined as:


 * $\forall s \in S: \map {\size f} s := \size {\map f s}$

where $\size {\map f s}$ denotes the extended absolute value function on $\overline \R$.

Absolute value thence is an instance of a pointwise operation on extended real-valued functions.

Since extended absolute value coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.

Also see

 * Definition:Pointwise Operation on Extended Real-Valued Functions for more operations on extended real-valued functions
 * Definition:Absolute Value of Mapping for the absolute value of more general mappings
 * Definition:Absolute Value of Real-Valued Function