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:


 * $\displaystyle \omega_f \left({A}\right) := \sup_{x, y \mathop \in A} \left\lvert{f \left({x}\right) - f \left({y}\right)}\right\rvert$

where the supremum is taken in the extended real numbers $\overline \R$.