Definition:Oscillation/Real Space/Oscillation at Point/Limit

Definition
Let $X$ be a set.

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

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

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:


 * $\displaystyle \omega_f \left({x; d}\right) := \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right); d}\right)$

where $\omega_f \left({\left({x - h \,.\,.\, x + h}\right); d}\right)$ denotes the oscillation of $f$ on $\left({x - h \,.\,.\, x + h}\right)$.

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.