Abstract functions with continuous differences and Namioka spaces

Let $G$ be a semigroup and a topological space. Let $X$ be an Abelian topological group. The right differences $\triangle _{h} \varphi$ of a function $\varphi : G \to X$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\triangle _{h} \varphi$ be continuous at the identity $e$ of $G$ for all $h$ in a neighbourhood $U$ of $e$. We give conditions on $X$ or range $\varphi$ under which $\varphi$ is continuous for any topological space $G$. We also seek conditions on $G$ under which we conclude that $\varphi$ is continuous at $e$ for arbitrary $X$. 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.