In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that 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 be a sequence of real numbers. The oscillation of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of :
The oscillation is zero if and only if the sequence converges. It is undefined if and are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.
Let be a real-valued function of a real variable. The oscillation of on an interval in its domain is the difference between the supremum and infimum of :
More generally, if is a function on a topological space (such as a metric space), then the oscillation of on an open set is
The oscillation of a function of a real variable at a point is defined as the limit as of the oscillation of on an -neighborhood of :
This is the same as the difference between the limit superior and limit inferior of the function at , provided the point is not excluded from the limits.
More generally, if is a real-valued function on a metric space, then the oscillation is
has oscillation ∞ at = 0, and oscillation 0 at other finite and at −∞ and +∞.
(the topologist's sine curve) has oscillation 2 at = 0, and 0 elsewhere.
has oscillation 0 at every finite , and 2 at −∞ and +∞.
or 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.
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In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Although the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.
Explores periodic functions, limits, compositions, and properties of even and odd functions.
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