Definition:Pointwise Inequality of Real-Valued Functions
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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