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

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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}).
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91--96
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Bibliogr. 10 poz.
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• [1] J. Aczél, S. Gołąb, Remarks on one-parameter subsemigroups of the affine group and their homo- and isomorphism, Aequationes Math. 4 (1970), 1-10.
• [2] A. Beck, A. A. Corson, A. B. Simon, The interior points of the product of two subsets of a locally compact group, Proc. Amer. Math. Soc. 9 (1958), 648-652.
• [3] J. Brzdęk, Subgroups of the group Zn and a generalization of the Gołąb-Schinzel functional equation, Aequationes Math. 43 (1992), 59-71.
• [4] J. Brzdęk, Some remarks on solutions of the functional equation f(x + f(x)ny) =tf(x)f(y), Publ. Math. Debrecen 43 (1993), 147-160.
• [5] J. Brzdęk, The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation, Ann. Polon. Math. 44, no. 3 (1996), 195-205.
• [6] J. Brzdęk, Bounded solutions of the Gołąb-Schinzel equation, Aequationes Math. 59 (2000), 248-254.
• [7] S. Gołąb, A. Schinzel, Surl'equation fonctionnelle f(x+f(x)y) = f(x)f(y), Publ. Math. Debrecen 6 (1959), 113-125.
• [8] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, PWN and Uniw. Śląski, Warszawa-Kraków-Katowice, 1985.
• [9] J. C. Oxtoby, Measure and Category, Springer Verlag, New York-Heidelberg-Berlin, 1971.
• [10] C. G. Popa, Sur l'equation fonctionnelle f(x+f(x)y) = f(x)f(y), Ann. Polon. Math. 17 (1965), 193-198.
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