nearcontinuous

Nearcontinuous, sometimes written as near-continuous, is a term used in real analysis and topology to describe a function that behaves continuously on a large subset of its domain. In common usage, a function f: X → Y is nearcontinuous if the set of points at which f is continuous is large in a topological sense, typically dense in X, and often required to be a dense G_delta. Equivalently, the set of discontinuities is meager (of first category) in X.

In metric spaces, the emphasis is usually on the function being continuous at a dense subset of the domain, with many references additionally requiring the continuity points to form a dense G_delta. This perspective aligns nearcontinuous functions with ideas from Baire category and the study of functions of Baire class 1, where the set of continuity points of such functions is known to be dense and G_delta.

Examples and related ideas: Thomae’s function, defined by f(x) = 0 for irrational x and f(p/q) = 1/q for rational x in lowest terms, is continuous at every irrational and discontinuous at every rational; since the irrationals form a dense G_delta, this function is nearcontinuous. It is not necessarily continuous everywhere, illustrating that nearcontinuity is a weaker property than global continuity. The concept is closely related to quasi-continuity and to the broader study of continuity properties within Baire category theory.

See also: Baire category, Baire class 1 functions, Thomae’s function, quasi-continuity, nearly open.