Real Analysis Exchange

On I-Asymmetry

Mariusz Strześniewski

Full-text: Open access


Sets of approximative asymmetry in the sense of category are introduced. The following theorem is proved. If $f : \mathbb{R}\to \mathbb{R}$ is a function, then the set of $\mathcal{I}$-asymmetry points of $f$ is of the type $F_{\sigma \delta \sigma}$ and is $\sigma$-well-porous. This illustrates the difference between measure and category. We give an example of a function with the set of $\mathcal{I}-$asymmetry points of the cardinality of the continuum.

Article information

Real Anal. Exchange, Volume 26, Number 2 (2000), 593-602.

First available in Project Euclid: 27 June 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A03: Foundations: limits and generalizations, elementary topology of the line 26E99: None of the above, but in this section

cluster set $\cal I$-density porosity asymmetry set


Strześniewski, Mariusz. On I -Asymmetry. Real Anal. Exchange 26 (2000), no. 2, 593--602.

Export citation


  • C. L. Belna, Cluster sets of arbitrary functions, Real Analysis Exchange, 1 (1978), No 1, 7–20.
  • L. Belowska, Resolution d'un probleme de M. Z. Zahorski sur les limites approximatives, Fund. Math. 48 (1960), 277–286.
  • E. P. Dolżenko, The boundary properties of arbitrary functions, Izv. Acad. Nauk SSSR, Ser. mat., 31 (1967), 3–14.
  • C. Goffman, On the approximate limits of real functions, Acta Sci. Math., 23 (1962), 76–78.
  • U. Hunter, Essential cluster sets, Trans. Amer. Math. Soc., 119 (1965), 350–388.
  • J. Jaskuła, On the set of points of the approximative assymetry, Ph. D. Thesis, University of Łódź, 1971.
  • J. Jędrzejewski, On the limit numbers of real functions, Fund. Math., 83 (1973/74), 269–281.
  • S. Kempisty, Sur les functions approximativement discontinues, Fund. Math., 6 (1924), 6–8.
  • M. Kulbacka, Sur l'ensemble des points de l'asymetrie approximative, Acta Sci. Math. Szeged, 21 (1960), 90–93.
  • J. S. Lipiński, Sur la discontinuite approximative et le derivee approximative, Colloq. Math. 10 (1963), 103–109.
  • E. Łazarow, On the Baire class of I-approximate derivatives, Proc. Amer. Math. Soc., 100 (1987), 669–674.
  • E. Łazarow, W. Wilczyński, I-approximate derivatives, Rad. Mat., 5 (1989), 15–27.
  • A. Matysiak, Sur les limites approximatives, Fund. Math., 48 (1960), 363–366.
  • W. Poreda, E. Wagner-Bojakowska, W. Wilczyński, A category analogue of the density topology, Fund. Math., 125 (1985), 167–173.
  • W. Poreda, E. Wagner-Bojakowska, W. Wilczyński, Remarks on I-density and I-approximately continuous functions, Comment. Fund. Math. Univ. Carolinae, 26 (1985), 553–564.
  • T. Świątkowski, On some generalization of the notion of assymetry of functions, Colloq. Math., 17 (1967), 77–91.
  • B. S. Thomson, Real Functions, Lect. Notes in Math. 1170 (1980), Springer Verlag.
  • L. Zajiček, Sets of $\partial$-porosity (q), \uCas. pro pest. mat., 101 (1976), 350–359.
  • L. Zajiček, On cluster sets of arbitrary functions, Fund. Math., 83 (1974), 197–217.
  • L. Zajiček, Alternative definitions of I-density topology, Acta Univ. Carolinae, 28 (1987), 57–61.
  • M. Strześniewski, A note on assymetry sets, Real Analysis Exchange, 14 (1988-89), 469–473.