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

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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:

$\displaystyle \omega_f \left({x}\right) := \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$

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