Generalized Cauchy equations on groups

It is well known (1], 3]) that any measurable solution of the Cauchy functional equationf(x+y)=f(x)+f(y) must actually be continuous. The same is true of some other functional equations likef(x+y)=f(x)f(y),f(x+y)f(x–y)=f(x) ² –f(y) ², etc. (cf. 1]). In this note we prove a general result of this type for functional equations on groups.