Introduction

The most famous functional equation is the Cauchy equation any solution of which is called additive. It is known that, for real valued functions defined on the real line, every Lebesgue measurable solution of (1.1) is of the form f (x) =cx for some constant c. Nonmeasurable solutions also exist but they are “wild”, being discontinuous everywhere and unbounded on every interval. A discussion of these facts may be found in Chapter 2 of the book by J. Aczél and J. Dhombres (1989).