# Definition:Oscillation/Oscillation on Set

< Definition:Oscillation(Redirected from Definition:Oscillation on Set)

## Definition

Let $X$ be a set.

Let $\left({Y, d}\right)$ 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 $\omega_f \left({A; d}\right)$, is defined as the diameter of $f \left({A}\right)$:

- $\displaystyle \omega_f \left({A; d}\right) := \operatorname{diam} \left({f \left({A}\right)}\right) = \sup_{x,y \mathop \in A} d \left({f \left({x}\right), f \left({y}\right)}\right)$

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 $\omega_f \left({A}\right)$.

Similarly, one would speak of the **oscillation of $f$ on $A$** in this case.