Approximate continuity

A generalization of the concept of continuity in which the ordinary limit is replaced by an approximate limit. A function is called approximately continuous at a point if

In the simplest case, is a real-valued function of the points of an -dimensional Euclidean space (in general it is a vector-valued function). The following theorems apply. 1) A real-valued function is Lebesgue-measurable on a set if and only if it is approximately continuous almost-everywhere on (the Stepanov–Denjoy theorem). 2) For any bounded Lebesgue-measurable function one has, at each point ,

where is the -dimensional Lebesgue measure, is an -dimensional non-degenerate segment containing , and is its diameter.

References

[1]	S. Saks, "Theory of the integral" , Hafner (1952) (Translated from French)

Comments

For other references see Approximate limit.