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 $\NN_x$ be the set of neighborhoods of $x$.

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


 * $\ds \map {\omega_f} x := \inf_{U \mathop \in \NN_x} \map {\omega_f} {U \cap X}$

where $\map {\omega_f} {U \cap X}$ denotes the oscillation of $f$ on $U \cap X$.