quasicontinuous

Quasicontinuous is a property of a function studied in topology and real analysis that lies between continuity and other regularity notions. It describes a function whose behavior can be controlled locally in a weaker sense than full continuity.

A function f: X → Y between topological spaces is quasicontinuous at a point x ∈ X if for every neighborhood V of x and every neighborhood W of f(x) there exists a nonempty open set U with x ∈ U ⊆ V such that f(U) ⊆ W. In words, no matter how small a neighborhood of x and how close one wishes f to stay to its value at x, one can find a small open region around x on which f remains within the prescribed neighborhood of f(x).

In metric spaces, an equivalent formulation is often used: f is quasicontinuous at x if for every ε > 0 and every neighborhood U of x there exists a nonempty open G ⊆ U with diam(f(G)) < ε. This emphasizes the possibility of obtaining arbitrarily small image variation on a suitably small open set around x.

Relation to other notions: every continuous function is quasicontinuous, but the converse need not hold. Quasicontinuity is therefore a weaker regularity condition that allows for more intricate local behavior than continuity permits, while still providing a form of local control over the image near a given point.

History and scope: the concept was introduced by A. Kempisty in 1961 in the study of real-valued functions. Since then, quasicontinuity has been explored in various settings within topology and analysis, contributing to the understanding of function regularity, pointwise limits, and related classifications. See also continuity, semicontinuity, and Baire class concepts.