Category:Definitions/Oscillation

This category contains definitions related to Oscillation.
Related results can be found in Category:Oscillation.

Real Space

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:

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

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

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 $\openint {x - h} {x + h} \cap X$.

Metric Space

Let $X$ be a set.

Let $\struct {Y, d}$ be a metric space.

Let $f: X \to Y$ be a mapping.

Oscillation on a Set

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.

Oscillation at a Point

Let $x \in X$.

Let $\tau$ be a topology on $X$, thus making $\struct {X, \tau}$ a topological space.

Denote with $\NN_x$ the set of neighborhoods of $x$ in $\struct {X, \tau}$.

The oscillation of $f$ at $x$ with respect to $d$, denoted by $\map {\omega_f} {x; d}$, is defined as:

$\ds \map {\omega_f} {x; d} := \inf_{U \mathop \in \NN_x} \map {\omega_f} {U; d}$

where $\map {\omega_f} {U; d}$ denotes the oscillation of $f$ on $U$.

The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\map {\omega_f} x$.

Similarly, one would speak of the oscillation of $f$ at $x$ in this case.