Definition:Oscillation/Metric Space/Set

Definition
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.