Oscillation (topology)

In the mathematics coming term of the oscillation in the topology before, one of the branches of mathematics . It also occurs in analysis and here in particular in integral calculus . Instead of the oscillation one also speaks of the fluctuation or the fluctuation range . The oscillation is used in the investigation of questions of continuity to mappings of topological spaces in metric spaces to measure the discontinuity of a mapping in a certain sense . Related to the concept of oscillation is that of the continuity module of mappings of metric spaces .

Oscillation of a sequence

The oscillation of a sequence (here the blue dots) is the difference between the Limes Superior and the Limes Inferior

Be a sequence of real numbers. The oscillation is defined as the difference between the limes superior and limes inferior of : ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \omega (a_{n})}$ ${\displaystyle (a_{n})_{n\in \mathbb {N} }}$

${\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}}$.

The oscillation of a sequence is zero if and only if the sequence converges. The oscillation is not defined if the limes superior and limes inferior are both equal or equal at the same time, i.e. if the sequence definitely diverges . ${\displaystyle +\infty }$ ${\displaystyle -\infty }$

Definitions, ways of speaking and writing

A topological space , a metric space and a map are given . ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle (Y,d_{Y})}$ ${\displaystyle f\colon X\to Y}$

Oscillation on a subset

For any non-empty subset , one understands by the oscillation from to or by the fluctuation from to the diameter of the image set with respect to the metric , i.e. the quantity which is defined as follows: ${\displaystyle U\subseteq X}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle f}$ ${\displaystyle U}$ ${\displaystyle f(U)}$ ${\displaystyle d_{Y}}$ ${\displaystyle {\Omega }_{f}(U)}$

${\displaystyle {\Omega }_{f}(U):=\operatorname {diam} _{d_{Y}}{f(U)}=\sup\{d_{Y}(f(a),f(b)):a,b\in U\}\in [0,\infty ]}$

In general, oscillation is also not excluded if - as is possible in the case of unlimited functions - no finite supremum exists. ${\displaystyle {\Omega }_{f}(U)=\infty }$

An often considered one case is that , where the amount metric , ie by the absolute value function given illustrating, while at the same time on limited is. Under these circumstances is ${\displaystyle Y=\mathbb {R} }$ ${\displaystyle d_{Y}}$ ${\displaystyle f}$ ${\displaystyle U}$

${\displaystyle {\Omega }_{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x)}$

With regard to the designation one finds also or ; sometimes, but rather in English-speaking sources . ${\displaystyle {\Omega }_{f}(U)}$ ${\displaystyle \omega (f,U)}$ ${\displaystyle \sigma (f,U)}$ ${\displaystyle \operatorname {osc} (f,U)}$

Oscillation in one point

In every neighborhood around the point p the function oscillates between f (a) and f (b) infinitely often. The oscillation of this function at point p is thus f (b) -f (a).

For a point one defines: ${\displaystyle x\in X}$

${\displaystyle {\omega }_{f}(x):=\inf\{{\Omega }_{f}(U):U\in {\mathcal {U}}_{x}\}\in [0,\infty ]}$

This quantity is called the oscillation of in the point or the oscillation of in (at) or also the point fluctuation of in (at) . By definition, the above infimum is formed over all environments in the environment filter. However, it is enough already, alone for determining that open environments within or even just any in environments contained neighborhood basis to consider. ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle {\mathcal {U}}_{x}}$ ${\displaystyle x}$ ${\displaystyle {\mathcal {U}}_{x}}$

Instead there is also the spelling or . In addition, if the context does not require any emphasis, the simple spelling or can be found. ${\displaystyle {\omega }_{f}(x)}$ ${\displaystyle {\omega }(x;f)}$ ${\displaystyle s_{f}(x)}$ ${\displaystyle f}$ ${\displaystyle {\omega }(x)}$ ${\displaystyle s(x)}$

Is the topological structure of also through a metric generated, so the area of the point filter has the - environments   ( ) than around the base and: ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle d_{X}}$ ${\displaystyle x\in X}$ ${\displaystyle \epsilon }$ ${\displaystyle U_{\epsilon }(x)}$ ${\displaystyle \epsilon >0}$

${\displaystyle {\omega }_{f}(x)=\lim _{\epsilon \to 0+}{{\Omega }_{f}(U_{\epsilon }(x))}}$

Studies of oscillation often occur - for example in integral calculus - in the event that the functions under consideration live on real intervals , i.e. are and are at the same time a restricted function . ${\displaystyle X=[a,b]\subset \mathbb {R} =Y}$ ${\displaystyle f:[a,b]\to \mathbb {R} }$

Since the open intervals of the form and also the closed intervals of the form form a surrounding basis for a point , one has: ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle U=(x-\epsilon ,x+\epsilon )\subseteq \mathbb {R} }$ ${\displaystyle U=[x-\epsilon ,x+\epsilon ]\subseteq \mathbb {R} }$   ${\displaystyle (\epsilon >0)}$

${\displaystyle \omega _{f}(x)=\lim _{\epsilon \to 0+}\Omega _{f}((x-\epsilon ,x+\epsilon )\cap [a,b])=\lim _{\epsilon \to 0+}\Omega _{f}([x-\epsilon ,x+\epsilon ]\cap [a,b])}$.

example

The function for positive ${\displaystyle f(x)=\sin \left({\tfrac {1}{x}}\right)}$ ${\displaystyle x}$

For the function

${\displaystyle f\colon \mathbb {R} \to \mathbb {R} \colon x\mapsto f(x)={\begin{cases}\sin \left({\tfrac {1}{x}}\right)&;x\neq 0\\0&;x=0\end{cases}}}$

is for and . ${\displaystyle \omega _{f}(x)=0}$ ${\displaystyle x\neq 0}$ ${\displaystyle \omega _{f}(0)=2}$

Results

1. The function is a continuous function above . ${\displaystyle {\omega }_{f}\colon x\mapsto {\omega }_{f}(x)\in [0,\infty ]}$
2. For a mapping from a topological to a metric space, continuity at one point means that the oscillation at this point is zero. In other words that means for is in continuous if and only if is. A mapping from a topological to a metric space is consequently continuous precisely if it does not have an oscillation greater than zero at any point. ${\displaystyle x\in X}$ ${\displaystyle f}$ ${\displaystyle x}$ ${\displaystyle {\omega }_{f}(x)=0}$
3. If one denotes with the set of discontinuities of and one inserts with , then applies ${\displaystyle \Delta (f)}$ ${\displaystyle f}$ ${\displaystyle \Delta _{N}(f)=\{x\in X:{\omega }_{f}(x)\geq {\tfrac {1}{N}}\}}$ ${\displaystyle N\in \mathbb {N} }$

   ${\displaystyle \Delta (f)=\bigcup _{N=1}^{\infty }\Delta _{N}(f)}$.

4. They are all closed sets and therefore there is always an F _σ -set . ${\displaystyle \Delta _{N}(f)}$ ${\displaystyle \Delta (f)}$
5. If a closed n-dimensional interval and a bounded real function, then Riemann-Darboux can be integrated if and only if they are all Jordan null sets . ${\displaystyle X=[{\vec {a}},{\vec {b}}]\subset {\mathbb {R} }^{n}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \Delta _{N}(f)}$

To the continuity module

→ Main article : Continuity modulus

The term continuity module, which is related to oscillation, was introduced by Henri Léon Lebesgue in 1910. The continuity module for a mapping between two metric spaces and and a given real number is the following size : ${\displaystyle f}$ ${\displaystyle (X,d_{X})}$ ${\displaystyle (Y,d_{Y})}$ ${\displaystyle \eta >0}$ ${\displaystyle \omega (f;\eta )}$

${\displaystyle \omega (f;\eta )=\sup\{d_{Y}(f(a),f(b)):a,b\in X\land d_{X}(a,b)\leq \eta \}\in [0,\infty ]}$

The continuity module has the following properties:

1. ${\displaystyle \omega (f;0)=0}$.
2. ${\displaystyle \eta \mapsto \omega (f;\eta )}$is monotonically increasing .
3. ${\displaystyle \eta \mapsto \omega (f;\eta )}$is subadditive .
4. ${\displaystyle \lim _{\eta \to 0}{\omega (f;\eta )}=0}$is equivalent to saying that is uniformly continuous . ${\displaystyle f}$

See also

• Limited variation

literature

• Hermann Athens, Jörn Bruhn (ed.): Lexicon of school mathematics and related areas . tape 4 : S-Z . Aulis Verlag, Cologne 1978, ISBN 3-7614-0242-2 .
• John J. Benedetto: Real Variable and Integration . BG Teubner Verlag, Stuttgart 1976, ISBN 3-519-02209-5 .
• Nicolas Bourbaki : Elements of Mathematics. General Topology. Part 2 (=  ADIWES International Series in Mathematics ). Addison-Wesley Publishing Company, Reading MA 1966.
• Harro Heuser : Textbook of Analysis. Part 1 (=  mathematical guidelines ). 16th revised edition. BG Teubner Verlag, Stuttgart 2006, ISBN 978-3-8351-0131-9 .
• Kazimierz Kuratowski : Topology . New ed., Revised and augmented. Volume 1. Academic Press, New York / London 1966 (translated from the French by J. Jaworowski).
• Serge Lang : Analysis . Ed .: Willi Jäger. Inter European Editions, Amsterdam 1977, ISBN 0-201-04152-9 (German translation by Bernd Wollring).
• John C. Oxtoby: Measure and Category. A Survey of the Analogies between Topological and Measure Spaces (  Graduate Texts in Mathematics . Volume 2 ). 2nd Edition. Springer Verlag, Berlin a. a. 1987, ISBN 3-540-90508-1 .
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . tape 4 . Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0436-3 .
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . tape 5 . Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0437-1 .
• Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970 ( MR0264581 ).

References and comments

1. ^ Lexicon of Mathematics in six volumes . tape 4 , p. 128-129 .
2. ^ Lexicon of school mathematics . tape 4 , p. 941-942 .
3. ^ H. Heuser: Textbook of Analysis . 2006, p. 241, 470-473 .
4. ^ JJ Benedetto: Real Variable and Integration . 1976, p. 24 .
5. ^ S. Lang: Analysis . 1977, p. 403 .
6. ^ JC Oxtoby: Measure and Category . 1987, p. 31 ff .
7. ^ S. Willard: General Topology . 1970, p. 177 .
8. ^ N. Bourbaki: Elements of Mathematics . 1966, p. 151 .
9. ^ K. Kuratowski: Topology . 1966, p. 208 .
10. ↑ In some places, an even more general situation is used as a basis. Then one looks at a non-empty subset and a mapping and then defines . For reasons of simplification, is then set at. See S. Willard: General Topology . 1970, p. ${\displaystyle (X,{\mathcal {O}})}$ ${\displaystyle A\subseteq X}$ ${\displaystyle f\colon A\to Y}$ ${\displaystyle {\Omega }_{f}(U):=\operatorname {diam_{d_{Y}}} {f(A\cap U)}=\sup\{d_{Y}(f(a),f(b)):a,b\in (A\cap U)\}\in [0,\infty ]}$ ${\displaystyle A\cap U=\emptyset }$   ${\displaystyle {\Omega }_{f}(U)=\infty }$ 177 .
11. ^ In N. Bourbaki: Elements of Mathematics . 1966, p. 151 . this quantity is also defined more generally for , and . ${\displaystyle A\subseteq X}$ ${\displaystyle f\colon A\to Y}$ ${\displaystyle x\in {\overline {A}}}$
12. ^ Lexicon of Mathematics in six volumes . tape 5 , p. 108 .