On the Kurepa and inhomogeneous Cauchy functional equations

TL;DR

This paper revisits the Kurepa and inhomogeneous Cauchy functional equations, providing new constructive proofs and estimates for solutions, thereby strengthening existing results and offering practical methods for constructing solutions.

Contribution

It offers new, constructive proofs for solutions of the Kurepa functional equation and estimates their modulus of continuity, enhancing previous theoretical results.

Findings

Solutions can be expressed as F(x,y)=f(x+y)-f(x)-f(y) with continuous or differentiable f

Constructive methods for building solutions are provided

Modulus of continuity for solutions is estimated

Abstract

It follows from de Bruijn's results that if a continuous or $k$-th order continuously differentiable function $F(x,y)$ is a solution of the Kurepa functional equation, then it can be expressed as $F(x,y)=f(x+y)-f(x)-f(y)$ with the continuous $f$ or the $k$-th order continuously differentiable $f$, respectively. These two facts strengthen the corresponding results of Kurepa and Erd\"{o}s. In this paper, we provide new and constructive proofs for these facts. In addition to practically useful recipes given here for construction of $f$, we also estimate its modulus of continuity.