Miscellanea About the Stability of Functional Equations

On the geometric concomitants

“…The solution F : R × R → R of the translation equation is of the form (5) if it is continuous with respect to the second variable for every x ∈ S and for which at least one of the functions F (., t) : R → R is continuous and F (x, 0) = x for x ∈ R ( [7], this form of F is proved in [8] if it is continuous). Notice that F is of this form too if it is Carathéodory, i.e.…”

On the geometric concomitant equation

“…If a solution F : R×R → R of the translation equation, such that F(x, 0) = x for x R, is continuous with respect to the second variable for every x R and for which at least one of the functions F(.,t) : R → R is continuous, then the function F has the form (6). In this case S n are open intervals and b n are homeomorphisms and the function F is continuous [10]. This solution F of the translation equation is continuous, too, if F is Carathéodory, i.e., the function F(x, .)…”