Oscillation (mathematics)

In mathematics, the concept of the oscillation occurs in the topology, one of the branches of mathematics. It also occurs in analysis and particularly in integral calculus. Instead of the oscillation is also called the variation or fluctuation. The oscillation is used in the study of continuity questions about pictures of topological spaces in metric spaces to measure the discontinuity of an image in a certain sense. Related to the concept of the oscillation is that of the continuity modulus of images of metric spaces.

Definitions, speaking and writing

Given a topological space is a metric space and a figure.

Oscillation on a subset

For any non- empty subset is understood to mean the oscillation of on or under the fluctuation of the diameter of the image set with respect to the metric, so that size which is defined as follows:

There also the oscillation is not generally excluded when - as possible in the case of unlimited features - no finite upper bound exists.

An often considered one is the case, that is, the amount of metrics, which is therefore given by the absolute value function, while at the same time is limited. Under these circumstances, is

Regarding the designation is also takes place or; sometimes, but rather in English sources.

Oscillation in a point

For a point is defined:

One calls this the size of the oscillation, the oscillation of points or in (for ), or the point of variation in ( in ). The above infimum is formed here by definition all environments in the vicinity filter. However, it is sufficient for the determination of which also have to consider only the open environments or even just within the environments contained in any neighborhood basis.

Instead, there is the case or. In addition, is provided out of context dependence of no emphasis needs to find the easy cases, or.

If the topological structure of also generated by a metric, the neighborhood filter of the point, the environments () as a basis of neighborhoods and we have:

Studies on the oscillation often occur - such as in the integral calculus - for the case that the functions considered living on real intervals, so and at the same time is a bounded function.

Since for a point, the open intervals of the form and also the closed intervals of the form form a neighborhood basis, one has:

Example

For the function with and for and.

Results

For modulus of continuity

The related with the oscillation term continuity module was introduced by Henri Léon Lebesgue in 1910. The modulus of continuity to a mapping between two metric spaces and and a given real number is the following size:

The modulus of continuity has the following properties:

Swell

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  • Harro Heuser: Textbook of analysis. Part 1 ( = Mathematical guides). 16, revised edition. BG Teubner Verlag, Stuttgart 2006, ISBN 978-3-8351-0131-9.
  • Kazimierz Kuratowski: Topology. Volume 1 New edition, revised and augmented. Translated from French by J. Jaworowski. Academic Press, New York - London 1966.
  • Serge Lang: Analysis. German translation by Bernd Wollring. Edited by Dr. Willi Jäger, Institute of Applied Mathematics, University of Heidelberg. Inter European Editions, Amsterdam 1977, ISBN 0-201-04152-9.
  • John C. Oxtoby: Measure and Category. A Survey of the Analogies in between Topological and Measure Spaces ( = Graduate Texts in Mathematics. 2). 2 edition. Springer Verlag, Berlin [u a ], 1987, ISBN 3-540-90508-1.
  • Guido Walz [ note]: Encyclopedia of mathematics in six volumes. Fourth band .. Spektrum Akademischer Verlag, Heidelberg - Berlin 2002, ISBN 38274-0436-3.
  • Guido Walz [ note]: Encyclopedia of mathematics in six volumes. Fifth band .. Spektrum Akademischer Verlag, Heidelberg - Berlin 2002, ISBN 38274-0437-1.
  • Stephen Willard: General Topology. Addison -Wesley, Reading, Massachusetts [u a ] 1970. MR0264581
Mathematical analysis Interval (mathematics)
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