nearcontinuity

Nearcontinuity is a mathematical concept that describes functions which exhibit properties similar to continuity but with certain relaxed conditions. Unlike strict continuity, which requires that the limit of a function as it approaches any point equals the function's value at that point, nearcontinuity allows for limited discontinuities or controlled deviations from this condition.

In formal terms, a function f is nearcontinuous if for every point x in its domain and every ε > 0, there exists a neighborhood around x where the function's values are within ε of f(x), except possibly at a set of points with measure zero or some other restricted set.

This concept is particularly useful in analysis and topology when dealing with functions that have isolated discontinuities or when working with approximations of continuous functions. Nearcontinuity preserves many useful properties of continuous functions while allowing for more flexibility in applications.

The study of nearcontinuity has connections to various areas of mathematics including real analysis, functional analysis, and measure theory. It provides a framework for understanding functions that are 'almost' continuous in a precise mathematical sense.