Definition:Oscillation/Metric Space/Point

Definition

Let $X$ be a set.

Let $\struct {Y, d}$ be a metric space.

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

Let $x \in X$.

Let $\tau$ be a topology on $X$, thus making $\struct {X, \tau}$ a topological space.

Denote with $\NN_x$ the set of neighborhoods of $x$ in $\struct {X, \tau}$.

The oscillation of $f$ at $x$ with respect to $d$, denoted by $\map {\omega_f} {x; d}$, is defined as:

$\ds \map {\omega_f} {x; d} := \inf_{U \mathop \in \NN_x} \map {\omega_f} {U; d}$

where $\map {\omega_f} {U; d}$ 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 $\map {\omega_f} x$.

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