Definition:Oscillation/Real Space

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Definition

Let $X$ and $Y$ be real sets.

Let $f: X \to Y$ be a real function.


Oscillation on a Set

Let $A \subseteq X$ be any non-empty subset $A$ of $X$.


The oscillation of $f$ on (or over) $A$ is defined as:

$\displaystyle \map {\omega_f} A := \sup_{x, y \mathop \in A} \size {\map f x - \map f y}$

where the supremum is taken in the extended real numbers $\overline \R$.


Oscillation at a Point

Let $x \in X$.


Definition 1

Let $\NN_x$ be the set of neighborhoods of $x$.


The oscillation of $f$ at $x$ is defined as:

$\ds \map {\omega_f} x := \inf_{U \mathop \in \NN_x} \map {\omega_f} {U \cap X}$

where $\map {\omega_f} {U \cap X}$ denotes the oscillation of $f$ on $U \cap X$.


Definition 2

The oscillation of $f$ at $x$ is defined as:

$\ds \map {\omega_f} x := \inf \set {\map {\omega_f} {\openint {x - \epsilon} {x + \epsilon} \cap X}: \epsilon \in \R_{>0} }$

where $\map {\omega_f} {\openint {x - \epsilon} {x + \epsilon} \cap X}$ denotes the oscillation of $f$ on $\openint {x - \epsilon} {x + \epsilon} \cap X$.


Definition 3

The oscillation of $f$ at $x$ is defined as:

$\ds \map {\omega_f} x := \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap X}$

where $\map {\omega_f} {\openint {x - h} {x + h} \cap X}$ denotes the oscillation of $f$ on $\openint {x - h} {x + h} \cap X$.