Real Analysis Exchange

Every real function is the sum of two extendable connectivity functions

Harvey Rosen

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Abstract

It is shown that an arbitrary fucntion \(f:\mathbb{R} \to \mathbb{R}\) can be written as the sum of two extendable connectivity functions.

Article information

Source
Real Anal. Exchange, Volume 21, Number 1 (1995), 299-303.

Dates
First available in Project Euclid: 3 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1341343245

Mathematical Reviews number (MathSciNet)
MR1377539

Zentralblatt MATH identifier
0847.26003

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}
Secondary: 54C08: Weak and generalized continuity

Keywords
continuity Peano derivative

Citation

Rosen, Harvey. Every real function is the sum of two extendable connectivity functions. Real Anal. Exchange 21 (1995), no. 1, 299--303. https://projecteuclid.org/euclid.rae/1341343245


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References

  • J. B. Brown, Totally discontinuous connectivity functions, Colloq. Math. 23 (1971), 53–60.
  • A. M. Bruckner and J. G. Ceder, On jumping functions by connected sets, Czech. Math. J. 22 (1972), 435–448.
  • H. Fast, Une remarque sur la propriete de Weierstrass, Colloq. Math. 7 (1959), 75–77.
  • R. G. Gibson, A property of Borel measurable functions and extendable functions, Real Analysis Exch. 13 (1987–88), 11–15.
  • R. G. Gibson, and F. Roush, Connectivity functions defined on $I^n$, Colloq. Math. 55 (1988), 41–44.
  • M. R. Hagan, Equivalence of connectivity maps and peripherally continuous transformations, Proc. Amer. Math. Soc. 17 (1966), 175–177.
  • K. R. Kellum, Sums and limits of almost continuous functions, Colloq. Math. 31 (1974), 125–128.
  • A. Lindenbaum, Sur quelques proprietes des fonctions de variable reelle, Ann. Soc. Polon. Math. 6 (1927), 129.
  • T. Natkaniec, Extendability and almost continuity, preprint.
  • D. Phillips, Real functions having graphs connected and dense in the plane, Fund. Math. 75 (1972), 47–49.
  • H. Rosen, Limits and sums of extendable connectivity functions, Real Analysis Exch. 20 (1994–95), 183–191.
  • H. Rosen, R. G. Gibson, and F. Roush, Extendable functions and almost continuous functions with a perfect road, Real Analysis Exch. 17 (1991–92), 248–257.
  • J. R. Stallings, Fixed point theorems for connectivity maps, Fund. Math. 47 (1959), 249–263.
  • G. T. Whyburn, Connectivity of peripherally continuous functions, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 1040–1041.