Abstract functions with continuous differences and Namioka spaces

Abstract

Let G G be a semigroup and a topological space. Let X X be an Abelian topological group. The right differences △ h φ \triangle _{h} \varphi of a function φ : G → X \varphi : G \to X are defined by △ h φ ( t ) = φ ( t h ) − φ ( t ) \triangle _{h}\varphi (t) = \varphi (th) - \varphi (t) for h , t ∈ G h,t \in G . Let △ h φ \triangle _{h} \varphi be continuous at the identity e e of G G for all h h in a neighbourhood U U of e e . We give conditions on X X or range φ \varphi under which φ \varphi is continuous for any topological space G G . We also seek conditions on G G under which we conclude that φ \varphi is continuous at e e for arbitrary X X . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.