Real Analysis Exchange

On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions

Zbigniew Grande

Full-text: Open access

Abstract

In this article we prove that if a function $f:X \to {\cal R}$ is the pointwise (discrete) [transfinite] limit of a sequence of real functions $f_n$ with closed graphs defined on complete separable metric space $X$ then $f$ is the pointwise (discrete) [transfinite] limit of a sequence of continuous functions. Moreover we show that each Lebesgue measurable function $f:{\cal R} \to {\cal R}$ is the discrete limit of a sequence of functions with closed graphs in the product topology $T_d\times T_e$, where $T_d$ denotes the density topology and $T_e$ the Euclidean topology.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 933-942.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571383

Mathematical Reviews number (MathSciNet)
MR1844409

Zentralblatt MATH identifier
1024.26003

Subjects
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} 54C30: Real-valued functions [See also 26-XX]

Keywords
Function with closed graph discrete convergence pointwise convergence transfinite convergence density topology

Citation

Grande, Zbigniew. On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions. Real Anal. Exchange 26 (2000), no. 2, 933--942. https://projecteuclid.org/euclid.rae/1214571383


Export citation

References

  • Baggs I.; Functions with a closed graphs, Proc. Amer. Math. Soc. 43 (1974), 439–442.
  • Borsík J., Doboš J. and Repický M.; Sums of quasicontinuous functions with closed graphs, Real Analysis Exchange 25 No. 2 (1999-2000), 679–690.
  • Bruckner A.M.; Differentiation of real functions, Lectures Notes in Math. 659, Springer-Verlag, Berlin 1978.
  • A. Császár and M. Laczkovich; Discrete and equal convergence, Studia Sci. Math. Hungar. 10 (1975), 463–472.
  • Davies R. O.; Approximate continuity implies measurability, Math. Proc. Camb. Philos. Soc. 73 (1973), 461–465.
  • Doboš J.; A note on the functions the graph of which are closed sets, Acta Math. Univ. Comenian. 40-41 (1982), 285–288.
  • Doboš J.; On the set of points of discontinuity for functions with closed graphs, Cas. pest. mat. 110 (1985), 60–68.
  • Grande Z.; Baire functions and their restrictions to special sets, Math. Slovaca 43 (1993), 447–453.
  • Kirchheim B.; Baire one star functions, Real Analysis Exchange 8 (1992//93), 385–389.
  • Kostyrko P.; A note on the functions with closed graphs, Cas. pest. mat. 94 (1969), pp. 202–205.
  • Lipiński J. S.; On transfinite sequences of mappings, Cas. pest. mat. 101 (1976), 153–158.
  • Sierpiński W.; Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math. 1 (1920), 132–141.
  • Tall F. D.; The density topology, Pacific J. Math. 62 (1976), pp.275-284.