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.

Elementary Properties
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 $x \in X$ $\map {\omega_f} x = 0$.


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


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