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)
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)