# Definition:Pointwise Inequality of Real-Valued Functions

Jump to navigation
Jump to search

## Definition

Let $S$ be a set.

Let $f, g: S \to \R$ be real-valued functions.

Then **pointwise inequality of $f$ and $g$**, denoted $f \le g$, is defined to hold if and only if:

- $\forall s \in S: f \left({s}\right) \le g \left({s}\right)$

where $\le$ denotes the usual ordering on the real numbers $\R$.

Thence **pointwise inequality** of real-valued functions is an instance of an induced relation on mappings.

## Also see

- Pointwise Inequality of Extended Real-Valued Functions, a similar concept for extended real-valued functions
- Pointwise Inequality, an abstraction replacing $\R$ by an arbitrary ordered set