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 $\mathcal N_x$ be the set of open subset neighborhoods of $x$.


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

$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U \cap X}\right)$

where $\omega_f \left({U \cap X}\right)$ denotes the oscillation of $f$ on $U \cap X$.


Definition 2

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

$\displaystyle \omega_f \left({x}\right) := \inf \left\{{\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right): \epsilon \in \R_{>0}}\right\}$

where $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X$.


Definition 3

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

$\displaystyle \omega_f \left({x}\right) := \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$

where $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - h \,.\,.\, x + h}\right) \cap X$.