Daróczy, Z.; Laczkovich, M. On functions taking the same value on many pairs of points. (English) Zbl 1170.26006 Real Anal. Exch. 33(2007-2008), No. 2, 385-394 (2008). Let \(I\subset\mathbb{R}\) be an interval. A function \(f: I\to\mathbb{R}\) is said to have the property \((M_p)\), where \(0< p< 1\), \(p\neq{1\over 2}\), if whenever \(x,y\in I\) and \(f(x)\neq f(y)\) then \(f(px+ (1- p)y)= f((1- p)x+ py)\). The authors show that if \(f\) is a measurable function with property \((M_p)\), then \(f\) is constant a.e. Also, if “measurable” is replaced by “has a point of continuity in \(I\)”, then \(f\) is constant apart from a countable set. The authors then apply this result to a certain functional equation. Reviewer: Khristo N. Boyadzhiev (Ada) Cited in 3 Documents MSC: 26B99 Functions of several variables 39B22 Functional equations for real functions Keywords:functional equations; measurability × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] A. M. Bruckner, Differentiation of real functions , · Zbl 0382.26002 [2] A. M. Bruckner, J. B. Bruckner, B. S. Thomson, Real Analysis , Prentice-Hall, 1997. · Zbl 0872.26001 [3] J. A. Clarkson, A property of derivatives , Bull. Amer. Math. Soc., 53 (1947), 124-125. · Zbl 0032.27102 · doi:10.1090/S0002-9904-1947-08757-7 [4] Z. Daróczy and Zs. Páles, Gauss-composition of means and the solution of the Matkowski-Sutô problem , Publ. Math. Debrecen, 61 (1-2) (2002), 157-218. · Zbl 1006.39020 [5] Z. Daróczy and Zs. Páles, On functional equations involving means , Publ. Math. Debrecen, 62 (3-4) (2003), 363-377. · Zbl 1026.39009 [6] A. Denjoy, Sur une proprieté des fonctions dérivées , Enseign. Math., 18 (1916), 320-328. · JFM 46.0381.05 [7] D. Głazowska, W. Jarczyk, J. Matkowski, Arithmetic mean as a linear combination of two quasi-arithmetic means , Publ. Math. Debrecen, 61 (3-4) (2002), 455-467. · Zbl 1012.26022 [8] A. Járai, Regularity Properties of Functional Equations in Several Variables , Springer, New York, 2005. [9] J. Matkowski, Invariant and complementary quasi-arithmetic means , Aequationes Math., 57 (1) (1999), 87-107. · Zbl 0930.26014 · doi:10.1007/s000100050072 [10] J. von Neumann, Ein system algebraisch unabhängiger Zahlen , Math. Ann., 99 (1928), 134-141. · JFM 54.0096.02 · doi:10.1007/BF01459089 [11] Problems of the 2007 Miklós Schweitzer Memorial Competition in Mathematics, http://www.math.u-szeged.hu/schw07. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.