Definition:Oscillation (Analysis)

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This page is about oscillation in the context of analysis. For other uses, see oscillation.

Definition

Real Space

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 $\overline \R$.


Metric Space

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.