Definition:Oscillation/Metric Space
Definition
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.
Elementary Properties
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With $X$ and $\struct {Y, d}$ as in the definitions above, the following hold:
- If $A \subseteq B$ are nonempty subsets of $X$ then $\map {\omega_f} A \le \map {\omega_f} B$.
- The infimum in the definition of $\map {\omega_f} x$ can be taken over the open neighborhoods as well and that definition would yield the same result.
- The oscillation satisfies the inequalities $0 \le \map {\omega_f} x \le \map {\omega_f} U \le \infty$ for any neighborhood $U$ of $x$ and all possibilities do occur for functions $f: \R \to \R$, for example.
- For all $r > 0$ the set $\set {x \in X: \map {\omega_f} x < r}$ is open.
- A function $f: X \to Y$ is continuous at $x \in X$ if and only if $\map {\omega_f} x = 0$.
- The set of discontinuities, $\map D f$, for a function $f: X \to Y$ can be written as a countable union of closed sets:
- $\ds \map D f = \bigcup_{n \mathop = 1}^\infty \set {x \in X: \map {\omega_f} x \ge \frac 1 n}$.