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

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

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

Let $x \in X$. Let a neighborhood of $x$ be a real set which contains (as a subset) an open set which itself contains (as an element) $x$.

Denote with $\mathcal N_x$ the set of 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 $U \cap X$.