On the Kurepa and inhomogeneous Cauchy functional equations

TL;DR

This work studies the inhomogeneous Cauchy (Kurepa) functional equation through the lens of regularity transfer: whenever a two-variable solution $F$ has a certain smoothness, it can be represented as $F(x,y)=f(x+y)-f(x)-f(y)$ with a univariate witness $f$ of corresponding regularity. Building on de Bruijn's difference-property framework, the authors give constructive proofs showing that if $F\in C^{k}(\mathbb{R}^{2})$, then there exists $f\in C^{k}(\mathbb{R})$ such that $F(x,y)=f(x+y)-f(x)-f(y)$, and similarly for $F\in C(\mathbb{R}^{2})$ to obtain $f\in C(\mathbb{R})$. They provide explicit recipes to construct $f$ from $F$ (and from $g$ via rational limits) and derive modulus-of-continuity bounds, notably $\omega(f;\delta;[-M,M]) \le 3\,\omega(F;\delta;[-M,M]^{2})$, enabling quantitative regularity control. The results yield constructive, quantitative connections between the smoothness of two-variable solutions and the regularity of the corresponding univariate function, strengthening classical Kurepa–Erdős conclusions.

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 Erdös. 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.