Definition:Pointwise Minimum of Mappings/Real-Valued Functions

Definition

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

Let $\min$ be the min operation on $\R$ (Ordering on 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:

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

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

Also see

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