Definition:Oscillation/Metric Space

Definition
Let $X$ be a set, and let $\left({Y, d}\right)$ 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 $\omega_f \left({A; d}\right)$, is defined as the diameter of $f \left({A}\right)$:


 * $\displaystyle \omega_f \left({A; d}\right) := \operatorname{diam} \left({f \left({A}\right)}\right) = \sup_{x,y \mathop \in A} d \left({f \left({x}\right), f \left({y}\right)}\right)$

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 $\omega_f \left({A}\right)$.

Similarly, one would speak of the oscillation of $f$ on $A$ in this case.

Oscillation at a Point
Let $x \in X$, and suppose that $X$ is a topological space.

Denote with $\mathcal N_x$ the set of neighborhoods of $x$.

Then the oscillation of $f$ at $x$ with respect to $d$, denoted by $\omega_f \left({x; d}\right)$, is defined as:


 * $\displaystyle \omega_f \left({x; d}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U; d}\right)$

where $\omega_f \left({U; d}\right)$ 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 $\omega_f \left({x}\right)$.

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

Elementary Properties
With $X$ and $(Y,d)$ as in the definitions above, the following hold:


 * If $A \subseteq B$ are nonempty subsets of $X$ then $\omega_f(A) \le \omega_f(B)$.


 * The infimum in the definition of $\omega_f(x)$ can be taken over the open neighborhoods as well and that definition would yield the same result.


 * The oscillation satisfies the inequalities $0 \le \omega_f(x) \le \omega_f(U) \le \infty$ for any neighborhood $U$ of $x$ and all posibilities do occur for e.g. functions $f: \R \to \R$.


 * For all $r>0$ the set $\left\{x \in X \mathrel{}\middle\vert\mathrel{} \omega_f(x) < r\right\}$ is open.


 * A function $f: X \to Y$ is continuous at $x \in X$ if and only if $\omega_f(x) = 0$.


 * The set of discontinuities, $D(f)$, for a function $f: X \to Y$ can be written as a countable union of closed sets:


 * $\displaystyle D(f) = \bigcup_{n \mathop = 1}^\infty \left\{x \in X \mathrel{}\middle\vert\mathrel{} \omega_f(x) \ge \tfrac1n\right\}$.