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Continuity of Lebesgue measurable solutions of a generalized Gołąb-Schinzel equation

Jabłońska E.

Demonstratio Mathematica

2006

Vol. 39, nr 1

91-96

Let k, n be positive integers and let f : Rn -> R be a solution of the functional equation f(x + f(x)ky)=f(x)f(y). We prove that, if there is a real positive a such that the set [x is an element of Rn : |f(x)| is an element of (0,a)} contains a subset of positive Lebesgue measure, then f is continuous. As a consequence of this we obtain that every Lebesgue measurable solution f : Rn -> R of the equation is continuous or equal zero almost everywhere (i.e. there is a set A C R of the Lebesgue measure zero with f(Rn \ A) = {0}).