Definition:Oscillation/Metric Space

Definition
Let $X$ be a set.

Let $\left({Y, d}\right)$ be a metric space.

Let $f: X \to Y$ be a mapping.

Oscillation at a Point
Let $x \in X$.

Definition 2
Let $X$ be a real set.

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 $\left({Y, d}\right)$ as in the definitions above, the following hold:


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


 * 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 $0 \le \omega_f \left({x}\right) \le \omega_f \left({U}\right) \le \infty$ for any neighborhood $U$ of $x$ and all possibilities do occur for e.g. functions $f: \R \to \R$.


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


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


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


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