Oscillation (mathematics)

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much a sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real valued function at a point, and oscillation of a function on an interval (or open set).

Let ${\displaystyle (a_{n})}$ be a sequence of real numbers. The oscillation ${\displaystyle \omega (a_{n})}$ of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of ${\displaystyle (a_{n})}$:

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