Equicontinuity

The gleichgradige continuity is a term from calculus and extends the notion of continuity of a function in a specific manner to function droves.

Definition

Be and metric spaces as well as a subset of the set of functions that map X to Y. The family of functions is called equicontinuous at the point when:

The family is called equicontinuous if it is equicontinuous at each point.

Each function in an equi- continuous family of functions is particularly steady.

In the event that the family of functions would be only continuous, could have a different value for each function of the family. " Continuity of the same degree " therefore means that the oscillation of the function values ​​is limited by the same number.

This term is used both in functional analysis over the set of Arzelà - Ascoli as compactness criterion application as well as in the theory of functions, for every holomorphic on a region of locally bounded family functions there is locally equicontinuous, that is, each point has a neighborhood on which the family is equicontinuous.