Real Analysis Exchange

Universally bad Darboux functions in the class of additive functions

Dariusz Banaszewski

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The main result: For every family \(\mathcal{G}\) of additive functions with \(\text{card }{\mathcal{G}}=2^\omega\) if the covering of the family of all level sets of functions from \(\mathcal{G}\) is equal to \(2^\omega\), then there exists an additive Darboux function \(f\) such that \(f+g\) is Darboux for no \(g\in\mathcal{G}\).

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Real Anal. Exchange, Volume 22, Number 1 (1996), 284-291.

First available in Project Euclid: 1 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27}
Secondary: 26A51: Convexity, generalizations

Darboux function additive function universally bad Darboux function maximal additive family


Banaszewski, Dariusz. Universally bad Darboux functions in the class of additive functions. Real Anal. Exchange 22 (1996), no. 1, 284--291.

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  • A. M. Bruckner and J. Ceder, On the sum of Darboux functions, Proc. Amer. Math. Soc. 51 (1975), 97–102.
  • A. L. Cauchy, Cours d'analyse de l'Ecole Polytechnique, 1. Analyse algébrique, V., Paris, 1821.
  • B. Kirchheim and T. Natkaniec, On universally bad Darboux functions, Real Analalysis Exch. 16 (1990–1991), 481–486.
  • P. Komjáth, A note on Darboux functions, Real Analalysis Exch. 18 (1992–93), 249–252.
  • M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality, PWN Warszawa–Kraków–Katowice 1985.
  • T. Radaković, Über Darbouxsche und stetige Funktionen, Monat. Math. Phys. 38 (1931), 117–122.
  • J. Steprans, Sums of Darboux and continuous functions, Fund. Math. 146 (1995), 107–120.
  • R. Švarc, On the range of values of the sum of a continuous and a Darboux function, Čas. Pest. Mat. 98 (1973), 178–180, 213.