Examples
- 1/x has oscillation ∞ at x = 0, and oscillation 0 at other finite x and at −∞ and +∞.
- sin (1/x) (the topologist's sine curve) has oscillation 2 at x = 0, and 0 elsewhere.
- sin x has oscillation 0 at every finite x, and 2 at −∞ and +∞.
- The sequence 1, −1, 1, −1, 1, −1, ... has oscillation 2.
In the last example the sequence is periodic, and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity.
Geometrically, the graph of an oscillating function on the real numbers follows some path in the xy-plane, without settling into ever-smaller regions. In well-behaved cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.
Read more about this topic: Oscillation (mathematics)
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