An application of nonstandard analysis to functional equations. (English) Zbl 0576.39008

Rašković, Miodrag

Publ. Inst. Math., Nouv. Sér. 37(51), 23-24 (1985).

The author shows that all measurable solutions f of the functional equation \(f(x+y)=g(f(x),f(y),x,y)\) where g is continuous, are continuous. A special case is the Cauchy functional equation \(f(x+y)=f(x)+f(y)\). The methodology is nonstandard analysis, (including Loeb measure).

Reviewer: Eugene Seneta (Sydney)

MSC:

39B99	Functional equations and inequalities
03H05	Nonstandard models in mathematics
26E35	Nonstandard analysis

Keywords:

continuous solution; measurable solutions; Cauchy functional equation; nonstandard analysis; Loeb measure