Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions

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

Let $\min$ be the min operation on $\overline{\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined).

Then the pointwise minimum of $f$ and $g$, denoted $\min \left({f, g}\right)$, is defined by:


 * $\min \left({f, g}\right): X \to \overline{\R}: \min \left({f, g}\right) \, \left({x}\right) := \min \left({f \left({x}\right), g \left({x}\right)}\right)$

Pointwise minimum thence is an instance of a pointwise operation on extended real-valued functions.

Since the ordering on $\overline{\R}$ coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.

Also see

 * Pointwise Maximum of Extended Real-Valued Functions, an analogous notion tied to the max operation
 * Pointwise Operation on Extended Real-Valued Functions for more operations on extended real-valued functions
 * Pointwise Minimum of Mappings for the pointwise minimum of more general mappings
 * Pointwise Minimum of Real-Valued Functions