Definition:Oscillation/Oscillation at Point

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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.

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.