Svetlana Velmar-Janković

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 be a Lebesgue measurable set, be a measurable function, and be a point where the Lebesgue density of is 1. The function is said to be approximately continuous at if and only if the approximate limit of at exists and equals .^[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 , a point is a Lebesgue point if it is a point of Lebesgue density 1 for and satisfies

where denotes the Lebesgue measure and represents the ball of radius centered at . Every Lebesgue point of a function is necessarily a point of approximate continuity.^[6] The converse relationship holds under additional constraints: when 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