\begin{definition}[Definition:Oscillation/Metric Space/Set]
Let $X$ be a set.
Let $\struct {Y, d}$ be a metric space.
Let $f: X \to Y$ be a mapping.
Let $A \subseteq X$ be any non-empty subset $A$ of $X$.
The '''oscillation of $f$ on (or over) $A$ with respect to $d$''', denoted $\map {\omega_f} {A; d}$, is defined as the diameter of $f \sqbrk A$:
:$\ds \map {\omega_f} {A; d} := \map \diam {f \sqbrk A} = \sup_{x, y \mathop \in A} \map d {\map f x, \map f y}$
where the supremum is taken in the extended real numbers $\overline \R$.
The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\map {\omega_f} A$.
Similarly, one would speak of the '''oscillation of $f$ on $A$''' in this case.
Category:Definitions/Metric Spaces
\end{definition}