Continuity of solutions of the translation equation

Abstract

Assuming that X is a metric space and F(0,∞)×X → X satisfies the translation equation F(s + t, x) = F(t, F(s, x)) for s, t (0,∞) and x X, we show that: 1. IfF is separately continuous, then it is continuous. 2. IfX is separable and F is Carathéodory, then F is continuous. We also show that if X is merely a topological, possibly separable, space, then a separately continuous mapping F : (0,∞) × X →X satisfying the above translation equation may fail to be continuous. © 2007 Birkhäuser Verlag AG.