Category:Definitions/Oscillation
This category contains definitions related to Oscillation.
Related results can be found in Category:Oscillation.
Real Space
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.
Pages in category "Definitions/Oscillation"
The following 12 pages are in this category, out of 12 total.
O
- Definition:Oscillation (Analysis)
- Definition:Oscillation at Point on Metric Space
- Definition:Oscillation of Real Function at Point
- Definition:Oscillation on Real Subset
- Definition:Oscillation on Set
- Definition:Oscillation/Metric Space
- Definition:Oscillation/Real Space
- Definition:Oscillation/Real Space/Oscillation at Point
- Definition:Oscillation/Real Space/Oscillation at Point/Epsilon
- Definition:Oscillation/Real Space/Oscillation at Point/Infimum
- Definition:Oscillation/Real Space/Oscillation at Point/Limit
- Definition:Oscillation/Real Space/Oscillation on Set