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
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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
,
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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.
Approximate continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_continuity&oldid=27390