Local Holder Continuity
The local Hölder continuity is a concept of mathematics that generalizes the Hölder continuity and thus also the Lipschitz continuity . It is named after Otto Hölder and is used, for example, in probability theory when formulating the Kolmogorow-Tschenzow theorem . This provides criteria as to when modifications of a stochastic process exist that are locally wood-continuous.
definition
Let two metric spaces and be given . An illustration
is called locally Hölder-continuous of order γ or locally Hölder-γ-continuous for short , if for each there is a really positive and a really positive number , so that for all with and the inequality
applies.
Examples
- Every Lipschitz continuous function with Lipschitz constant is locally Hölder continuous with exponent and
- Every Hölder-continuous mapping with constant and exponent is also locally Hölder-continuous with constant and exponent .
properties
- If a real-valued function of a real variable is locally Hölder-continuous with an exponent , then also locally Hölder-continuous for every exponent with .
- If the domain of definition is compact, then Hölder continuity follows from local Hölder continuity. In general, this conclusion is wrong.
literature
- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 468-469 , doi : 10.1007 / 978-3-642-36018-3 .