Approximately continuous function

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.^[1] This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.^[2]

Let ${\displaystyle E\subseteq \mathbb {R} ^{n}}${\displaystyle E\subseteq \mathbb {R} ^{n}} be a Lebesgue measurable set, ${\displaystyle f\colon E\to \mathbb {R} ^{k}}${\displaystyle f\colon E\to \mathbb {R} ^{k}} be a measurable function, and ${\displaystyle x_{0}\in E}${\displaystyle x_{0}\in E} be a point where the Lebesgue density of ${\displaystyle E}${\displaystyle E} is 1. The function ${\displaystyle f}${\displaystyle f} is said to be approximately continuous at ${\displaystyle x_{0}}${\displaystyle x_{0}} if and only if the approximate limit of ${\displaystyle f}${\displaystyle f} at ${\displaystyle x_{0}}${\displaystyle x_{0}} exists and equals ${\displaystyle f(x_{0})}${\displaystyle f(x_{0})}.^[3]

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.^[4] 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. ^[5]

Approximately continuous functions are intimately connected to Lebesgue points. For a function ${\displaystyle f\in L^{1}(E)}${\displaystyle f\in L^{1}(E)}, a point ${\displaystyle x_{0}}${\displaystyle x_{0}} is a Lebesgue point if it is a point of Lebesgue density 1 for ${\displaystyle E}${\displaystyle E} and satisfies

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

where ${\displaystyle \lambda }${\displaystyle \lambda } denotes the Lebesgue measure and ${\displaystyle B_{r}(x_{0})}${\displaystyle B_{r}(x_{0})} represents the ball of radius ${\displaystyle r}${\displaystyle r} centered at ${\displaystyle x_{0}}${\displaystyle x_{0}}. Every Lebesgue point of a function is necessarily a point of approximate continuity.^[6] The converse relationship holds under additional constraints: when ${\displaystyle f}${\displaystyle f} is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.^[7]

• Approximate limit
• Density point
• Density topology (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

• Theory of continuous functions
• Calculus
• Real analysis
• Mathematical analysis
• Measure theory
• Types of functions

• Articles with short description
• Short description matches Wikidata
• Articles lacking page references from January 2025
• Wikipedia articles that are too technical from May 2025
• All articles that are too technical
• Articles with multiple maintenance issues