On a Problem of Roger Cuculiere

Abstract

In the February 2008 issue of  The American Mathematical Monthly (115, Problems and Solutions, p. 166) the following question was proposed by Roger Cuculiere:

Find all nondecreasing functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x + f(y)) = f(f(x)) + f(y)$ for all real $x$ an $y$ (Problem 11345).
In the present paper we establish the general Lebesgue measurable solution, monotonic solutions as well as a description of the general solution of the functional equation in question.