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

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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