Definition:Oscillation/Real Space/Oscillation on Set

Definition

Let $X$ and $Y$ be real sets.

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 $\overline \R$.