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

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:

$\ds \map {\omega_f} x := \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} \cap X}$

where $\map {\omega_f} {\openint {x - h} {x + h} \cap X}$ denotes the oscillation of $f$ on $\openint {x - h} {x + h} \cap X$.

Sources

• 2010: J.N. Sharma and A.R. Vasishta: Mathematical Analysis-II: Chapter $6$, $\S 7$: Saltus at a point