Definition:Oscillation/Metric Space

Definition
Suppose $X$ is a set and $(Y,d)$ is a metric space. For any nonempty subset $A$ of $X$ the oscillation of $f$ on (or over) $A$ with respect to $d$ and denoted by $\omega_f(A;d)$ is defined as:


 * $\displaystyle \omega_f(A;d) = \sup_{x,y\in A} d(x,y),$

i.e. the diameter of $f(A)$. The metric is often omitted if it's clear from context which metric one is working with in which case one would write $\omega_f(A)$.

Suppose furthermore that $X$ is a space and $\mathcal N_x$ is the set of neighborhoods of a point $x \in X$, then the oscillation of $f$ at $x$ with respect to $d$ is denoted by $\omega_f(x;d)$ and it's defined as:


 * $\displaystyle \omega_f(x;d) = \inf_{U \in \mathcal N_x} \omega_f(U;d).$

As for oscillation over a set, the metric is often omitted when it's clear from context which metric one is working with.

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


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


 * 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 $0 \leq \omega_f(x) \leq \omega_f(U) \leq \infty$ for any neighborhood $U$ of $x$ and all posibilities do occur for e.g. functions $f: \mathbb R \to \mathbb R$.


 * 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:


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