What is Dubikajtis Theorem?

Question

Recently I was studying functional equations from the book Functional equations and how to solve them by G.Small. In which for Problem 24 of Chapter 3 the author has stated (in solution) that the problem can ask be solved by Dubikajtis theorem, but when I was searching in internet the theorem was some kind of stuff related to mathematical logic, but how we can we use it here or is there any such theorem ?

Answer 1

The reference you are looking for is the following paper:

Lech Dubikajtis, Sur certaines équations fonctionnelles vérifiées par la fonction $\varphi(x)=x^{-1}$ , Annales Polonici Mathematici 22 (1969), 199-205.

He shows that $\varphi(x)=x^{-1}$ is the unique continuous function which satisfies this system of functional equations for $x>0$, $$\begin{cases} f(f(x))=x\\ f(x+1)=\frac{f(x)}{f(x)+1} \end{cases} $$ The pdf is free to download.