\begin{definition}[Definition:Oscillation/Metric Space/Point]
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.
Category:Definitions/Metric Spaces
\end{definition}