Uniform continuity
The uniform continuity is a term from analysis and extends the concept of the continuity of a function in a special way to function families .
definition
Be and metric spaces as well as a subset of the set of functions to reproduce. The family of functions is called uniformly continuous at the point if:
The family is called equally continuous if it is equally continuous in every point .
Many authors use the term uniform continuity synonymously with uniformly uniform continuity .
In particular, each function in an equally continuous family of functions is continuous.
In the event that the family of functions were only continuous, each function of the family could have a different value. “Continuity of the same degree” means that the fluctuation of the function values is limited by the same number.
This term is used both in functional analysis using the Arzelà-Ascoli theorem as a compactness criterion and in function theory , because every family of holomorphic functions locally restricted in a domain is locally uniformly continuous there, i.e. every point has a neighborhood on which the family is equally steady.
See also
Individual evidence
- ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture. Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , paragraph 5.8.
- ^ R. Meise, D. Vogt: Introduction to functional analysis. Vieweg, 1992, ISBN 3-528-07262-8 , sentence 4.12.
- ↑ Wolfgang Fischer, Ingo Lieb: Function theory. Friedr. Vieweg & Sohn, 1980, ISBN 3-528-07247-4 , sentence IX, 6.3.