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: ${\displaystyle (X,d_{X})}$ ${\displaystyle (Y,d_{Y})}$ ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle F}$ ${\displaystyle x\in X}$

${\displaystyle \forall \varepsilon >0:\quad \exists \delta >0:\quad \forall f\in F:\quad \forall x'\in X:\quad d_{X}(x,x')\leq \delta \ \Rightarrow \ d_{Y}\left(f(x),f(x')\right)\leq \varepsilon .}$

The family is called equally continuous if it is equally continuous in every point . ${\displaystyle F}$ ${\displaystyle x\in X}$

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. ${\displaystyle F}$ ${\displaystyle \delta }$

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

• Uniform uniformity

Individual evidence

1. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture. Bibliographisches Institut Mannheim (1978), ISBN 3-411-00121-6 , paragraph 5.8.
2. ^ R. Meise, D. Vogt: Introduction to functional analysis. Vieweg, 1992, ISBN 3-528-07262-8 , sentence 4.12.
3. ↑ Wolfgang Fischer, Ingo Lieb: Function theory. Friedr. Vieweg & Sohn, 1980, ISBN 3-528-07247-4 , sentence IX, 6.3.