A class of mappings containing all continuous and all semiconnected mappings

Abstract

A function f : X → Y f:X \to Y is called s-continuous if for each x ∈ X x \in X and each open set V containing f ( x ) f(x) and having connected complement there is an open set U containing x such that f ( U ) ⊂ V f(U) \subset V . In this paper basic properties of s-continuous functions are studied; conditions on domain and/or range implying continuity of s-continuous functions are obtained which generalize recent theorems of Jones, Lee and Long on semiconnected functions. Improvements of recent results of Hagan, Kohli and Long concerning the continuity of certain connected functions follow as a consequence. Also characterizations of semilocally connected spaces in terms of s-continuous functions are obtained.