Definition:Oscillation/Real Space/Oscillation on Set
< Definition:Oscillation | Real Space(Redirected from Definition:Oscillation on Real Subset)
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Definition
Let $f: X \to Y$ be a real function.
Let $A \subseteq X$ be any non-empty subset $A$ of $X$.
The oscillation of $f$ on (or over) $A$ is defined as:
- $\ds \map {\omega_f} A := \sup_{x, y \mathop \in A} \size {\map f x - \map f y}$
where the supremum is taken in the extended real numbers $\overline \R$.