Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Approximately continuous function

Mathematical concept in measure theory

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated April 13, 2026

In mathematics, particularly in mathematical analysis and measure theory, an approximately continuous function is a concept that generalizes the notion of continuous functions by replacing the ordinary limit with an approximate limit. This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.

01Definition

Let E\subseteq \mathbb {R} ^{n} be a Lebesgue measurable set, f\colon E\to \mathbb {R} ^{k} be a measurable function, 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 of f at x_{0} exists and equals f(x_{0}).

02Properties

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 it is approximately continuous almost everywhere.

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 _{r\downarrow 0}{\frac {1}{\lambda (B_{r}(x_{0}))}}\int _{E\cap B_{r}(x_{0})}|f(x)-f(x_{0})|\,dx=0

where \lambda denotes the Lebesgue measure 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, its points of approximate continuity coincide with its Lebesgue points.

Watch videos about Approximately continuous functionExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Approximately continuous function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

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 R k {\displaystyle \mathbb {R} ^{k}} has an approximate limit y at a point x if there exists a set F that has density 1 at the point such that if xn is a sequence in F that converges towards x then f converges towards y.

Lebesgue's density theorem

In mathematics, Lebesgue's density theorem states that for any Lebesgue measurable set A ⊆ R n {\displaystyle A\subseteq \mathbb {R} ^{n}} , the "density" of A {\displaystyle A} is 0 or 1 at almost every point in R n {\displaystyle \mathbb {R} ^{n}} . Additionally, the "density" of A {\displaystyle A} is 1 at almost every point of A {\displaystyle A} .

Density topology

In mathematics, the density topology on the real numbers is a topology on the real line that is different, but in some ways analogous, to the usual topology. It is sometimes used in real analysis to express or relate properties of the Lebesgue measure in topological terms.