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:

$\ds \map D f = \bigcup_{n \mathop = 1}^\infty \set {x \in X: \map {\omega_f} x \ge \frac 1 n}$.