Definition:Oscillation

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Definition

Let $X$ be a set.

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 $x \in X$.

Let $X$ be a topological space.

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


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