Real Analysis Exchange

Additive Sierpiński-Zygmund functions.

Tomasz Natkaniec and Harvey Rosen

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In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the class of additive Sierpiński-Zygmund (SZ) functions. In particular, we give an example of an additive Sierpiński-Zygmund (SZ) injection $f : \mathbb{R} \to\mathbb{R}$ such that $f^{-1}$ is not an SZ function. Under the assumption that $\mathbb{R}$ cannot be covered by less than $\mathfrak{c}$-many meager sets we give examples of an additive SZ bijection $f : \mathbb{R} \to\mathbb{R}$ such that $f^{-1}$ is not SZ and of an additive injection $f : \mathbb{R} \to\mathbb{R}$ such that both $f$ and $f^{-1}$ are SZ.

Article information

Real Anal. Exchange, Volume 31, Number 1 (2005), 253-270.

First available in Project Euclid: 5 June 2006

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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: 03E50: Continuum hypothesis and Martin's axiom [See also 03E57]

additive function Sierpi{\'n}ski-Zygmund function Darboux like function almost continuous functions connectivity functions functions with perfect road peripherally continuous functions CIVP-functions SCIVP-functions


Natkaniec, Tomasz; Rosen, Harvey. Additive Sierpiński-Zygmund functions. Real Anal. Exchange 31 (2005), no. 1, 253--270.

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  • M. Balcerzak, K. Ciesielski and T. Natkaniec, Sierpiński-Zygmund Functions that are Darboux, Almost Continuous, or Have a Perfect Road, Arch. Math. Logic, 37 (1997), 29–35.
  • A. M. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. Vol. 659, Springer-Verlag, Berlin, 1978.
  • K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Student Texts, 39, Cambridge Univ. Press, 1997.
  • K. Ciesielski, Some Additive Darboux Like Functions, J. App. Anal., 4 (1997), 29–35.
  • K. Ciesielski, J. Jastrzębski, Darboux Like Functions within the Classes of Baire One, Baire Two, and Additive Functions, Topology Appl., 103 (2000), 203–219.
  • K. Ciesielski, T. Natkaniec, Algebraic Properties of the Class of Sierpiński-Zygmund Functions, Topology Appl., 79 (1997), 75–99.
  • K. Ciesielski, T. Natkaniec, On Sierpiński-Zygmund Bijections and Their Inverses, Topology Proc., 22 (1997), 155–164.
  • K. Ciesielski, J. Pawlikowski, The Covering Property Axiom. A Combinatorial Core of the Iterated Perfect Set Model, Cambridge Tracts in Mathematics, 164, Cambridge Univ. Press, Cambridge, 2004.
  • K. Ciesielski, A. Rosłanowski, Two Examples Concerning Almost Continuous Functions, Topology Appl., 103 (2000), 187–202.
  • U. B. Darji, A Sierpiński-Zygmund Function Which Has a Perfect Road at Each Point, Colloq. Math., 64 (1993), 159–162.
  • R. Gibson, T. Natkaniec, Darboux Like Functions, Real Anal. Exchange, 22(2) (1996–97), 492–533.
  • R. Gibson, T. Natkaniec, Darboux Like Functions. Old Problems and New Results, Real Anal. Exchange, 24(2) (1998–99), 487–496.
  • A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.
  • 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. Natkaniec, H. Rosen, An Example of an Additive Almost Continuous Sierpiński-Zygmund Function, Real Anal. Exchange, 30 (2004–05), 261–266.
  • K. Płotka, Sum of Sierpiński-Zygmund and Darboux Like Functions, Topology Appl., 122(3) (2002), 547–564.
  • W. Sierpiński, A. Zygmund, Sur une Fonction qui est Discontinue sur Tout Ensemble de Puissance du Continu, Fund. Math., 4 (1923), 316–318.