\begin{definition}[Definition:Oscillation/Real Space/Oscillation on Set]
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 $\overline \R$.
Category:Definitions/Oscillation
\end{definition}