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

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

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

Let $x \in X$.

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


 * $\displaystyle \omega_f \left({x}\right) := \inf \left\{{\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right): \epsilon \in \R_{>0}}\right\}$

where $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X$.