Definition:Oscillation/Real Space/Oscillation at Point/Epsilon
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Definition
Let $f: X \to Y$ be a real function.
Let $x \in X$.
The oscillation of $f$ at $x$ is defined as:
- $\ds \map {\omega_f} x := \inf \set {\map {\omega_f} {\openint {x - \epsilon} {x + \epsilon} \cap X}: \epsilon \in \R_{>0} }$
where $\map {\omega_f} {\openint {x - \epsilon} {x + \epsilon} \cap X}$ denotes the oscillation of $f$ on $\openint {x - \epsilon} {x + \epsilon} \cap X$.
Sources
- 2010: J.N. Sharma and A.R. Vasishta: Mathematical Analysis-II: Chapter $6$, $\S 7$: Saltus at a point