On continuous on rays solutions of a composite-type equation

Abstract

Abstract. Let X be a real linear space. We characterize solutions f, g : X → R of the equation f (x + g(x)y) = f (x)f (y), where f is continuous on rays. Our result refers to papers by

General regular variation, Popa groups and quantifier weakening

“…Care is needed in identifying locally SW sets: Matoȗsková and Zelený [MatZ] show that in any non-locally compact abelian Polish group there are closed non-Haar null sets A, B such that A + B has empty interior. Recently, Jabłońska [Jab4] has shown that likewise in any non-locally compact abelian Polish group there are closed non-Haar meager sets A, B such that A + B has empty interior. 2.…”

Functional equations of the Goła̧b-Schinzel type on a cone

“…In a series of papers [7], [9] and [12]- [14] the following generalization of the Gołab-Schinzel equation…”

Continuous on Rays Solutions of a Goła̧b–schinzel Type Equation

“…In the recent papers [7] and [8], Jabłońska considered the solutions of (1.1) in the real case under the assumption that f is continuous and continuous on rays, respectively. The main idea of the considerations presented in [7] and [8] is to prove that if the pair ( f, g) satisfies (1.1) and f is nonconstant then the continuity or continuity on rays of f implies the same property for g. Then, in order to determine the solutions of (1.1), it is enough to apply the results of [4] and [6], respectively.…”

“…The aim of this paper is to present a significantly shorter and simpler proof of the main results in [7] and [8] which works also in the complex case.…”