Approximate Continuity
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
and
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, an approximately continuous function is a concept that generalizes the notion of
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s by replacing the ordinary limit with an
approximate limit In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables. A function ''f'' on \mathbb^k has an approximate limit ''y'' at a point ''x'' if there exists a set ''F'' that h ...
. This generalization provides insights into
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s with applications in real analysis and geometric measure theory.


Definition

Let E \subseteq \mathbb^n be a Lebesgue measurable set, f\colon E \to \mathbb^k be a
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
, and x_0 \in E be a point where the Lebesgue density of E is 1. The function f is said to be ''approximately continuous'' at x_0 if and only if the
approximate limit In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables. A function ''f'' on \mathbb^k has an approximate limit ''y'' at a point ''x'' if there exists a set ''F'' that h ...
of f at x_0 exists and equals f(x_0).


Properties

A fundamental result in the theory of approximately continuous functions is derived from Lusin's theorem, which states that every measurable function is approximately continuous at almost every point of its domain. The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The Stepanov-Denjoy theorem provides a remarkable characterization:
Stepanov-Denjoy theorem: A function is measurable
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
it is approximately continuous
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.
Approximately continuous functions are intimately connected to Lebesgue points. For a function f \in L^1(E), a point x_0 is a Lebesgue point if it is a point of Lebesgue density 1 for E and satisfies :\lim_ \frac \int_ , f(x)-f(x_0), \, dx = 0 where \lambda denotes the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
and B_r(x_0) represents the ball of radius r centered at x_0. Every Lebesgue point of a function is necessarily a point of approximate continuity. The converse relationship holds under additional constraints: when f is
essentially bounded In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
, its points of approximate continuity coincide with its Lebesgue points.


See also

*
Approximate limit In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables. A function ''f'' on \mathbb^k has an approximate limit ''y'' at a point ''x'' if there exists a set ''F'' that h ...
*
Density point In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A\subset \R^n, the "density" of ''A'' is 0 or 1 at almost every point in \R^n. Additionally, the "density" of ''A'' is 1 at almost every point in ''A''. Intu ...
*
Density topology In mathematics, the density topology on the real numbers is a topology on the real line that is different (strictly finer), but in some ways analogous, to the usual topology. It is sometimes used in real analysis to express or relate properties of ...
(which serves to describe approximately continuous functions in a different way, as continuous functions for a different topology) * Lebesgue point * Lusin's theorem *
Measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...


References

{{reflist Theory of continuous functions Calculus Real analysis Mathematical analysis Measure theory Types of functions