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

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Let $X$ and $Y$ be real sets.

Let $f: X \to Y$ be a real function.

Let $x \in X$.

Let $\mathcal N_x$ be the set of open subset neighborhoods of $x$.

The oscillation of $f$ at $x$ is defined as:

$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U \cap X}\right)$

where $\omega_f \left({U \cap X}\right)$ denotes the oscillation of $f$ on $U \cap X$.