The authors in this study, firstly, identified the rough convergence and presented the set of rough limit points of a function defined on discrete countable amenable semigroups (DCASG) with some characteristics such as convexity, closedness and boundedness. Then, they introduced rough Cauchy sequence and also, examined the relations between rough Cau-chy sequences and rough convergence of functions defined on discrete countable amenable semigroups.