On Darboux solutions of the Euler equation (Q1121466)

Consider the Euler functional equation \(f(x+f(x))=f(x)\) for every \(x\in {\mathbb{R}}\). The author proves that there exists a non-constant Darboux function f (i.e., f maps connected sets onto connected sets) which is solution of Euler's equation on the whole line. This shows that a result in \textit{M. Kuczma's} book [Functional Equations in a single variable (1968; Zbl 0196.164) pp. 286-287] is false.