Real Analysis Exchange

On characterizing extendable connectivity functions by associated sets

Harvey Rosen

Full-text: Open access

Abstract

We answer two questions asked by Rosen in 1994. We show that the class of extendable connectivity functions from \(I\) into \(I\) (\(I = [0,1]\)) cannot be characterized in terms of associated sets, and we show that one of Jones’ functions obeying \(f(x+y) = f(x) + f(y)\) is an example of an almost continuous function from \(\mathbb{R}\) into \(\mathbb{R}\) which is not the uniform limit of any sequence of extendable connectivity functions.

Article information

Source
Real Anal. Exchange, Volume 22, Number 1 (1996), 279-283.

Dates
First available in Project Euclid: 1 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1338515220

Mathematical Reviews number (MathSciNet)
MR1433613

Zentralblatt MATH identifier
0879.26012

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} 54C08: Weak and generalized continuity

Keywords
extendable connectivity function associated set almost continuous function uniform limit

Citation

Rosen, Harvey. On characterizing extendable connectivity functions by associated sets. Real Anal. Exchange 22 (1996), no. 1, 279--283. https://projecteuclid.org/euclid.rae/1338515220


Export citation

References

  • A. M. Bruckner, On characterizing classes of functions in terms of associated sets, Canad. Math. Bull. 10 (1967), 227–231.
  • K. Ciesielski and I. Reclaw, Cardinal invariants concerning extendable and peripherally continuous functions, Real Analysis Exch., 21, no. 2 (1995–96), 459–472
  • B. Cristian and I. Tevy, On characterizing connected functions, Real Anal. Exchange 6 (1980-81), 203–207.
  • R. G. Gibson and F. Roush, Concerning the extension of connectivity functions, Top. Proc. 10 (1985), 75–82.
  • R. G. Gibson and F. Roush, The uniform limit of connectivity functions, Real Anal. Exchange 11 (1985-86), no. 1, 254–259.
  • J. Jastrzebski, An answer to a question of R. G. Gibson and F. Roush, Real. Anal. Exchange 15 (1989-90), 340–341.
  • F. B. Jones, Connected and disconnected plane sets and the functional equation $f(x) + f(y) = f(x+y)$, Bull. Amer. Math. Soc. 48 (1942), 115–120.
  • K. R. Kellum, Sums and limits of almost continuous functions, Colloq. Math. 31 (1974), 125–128.
  • K. R. Kellum, Almost continuity and connectivity – sometimes it's as easy to prove a stronger result, Real Anal. Exchange 8 (1982-1983), no. 1, 244–252.
  • T. Natkaniec, Extendability and almost continuity, Real Anal. Exchange 21 (1995-96), no. 1, 349–355.
  • H. Rosen, Limits and sums of extendable connectivity functions, Real Anal. Exchange 20 (1994-95), no. 1, 183–191.
  • H. Rosen, Every real function is the sum of two extendable connectivity functions, Real Anal. Exchange 21 (1995-96), no. 1, 299–303.