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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Cardinal invariants concerning
bounded families of extendable
and almost continuous functions

Authors: Krzysztof Ciesielski and Aleksander Maliszewski
Journal: Proc. Amer. Math. Soc. 126 (1998), 471-479
MSC (1991): Primary 26A21; Secondary 54C08
MathSciNet review: 1422855
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Abstract: In this paper we introduce and examine a cardinal invariant $\operatorname{A}_{\textup{b}}$ closely connected to the addition of bounded functions from $\mathbb{R}$ to $\mathbb{R}$. It is analogous to the invariant $\operatorname{A}$ defined earlier for arbitrary functions by T. Natkaniec. In particular, it is proved that each bounded function can be written as the sum of two bounded almost continuous functions, and an example is given that there is a bounded function which cannot be expressed as the sum of two bounded extendable functions.

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  • 1. J. B. Brown, P. D. Humke, and M. Laczkovich, Measurable Darboux functions, Proc. Amer. Math. Soc. 102(3) (1988), 603-612. MR 89b:26003
  • 2. A. M. Bruckner, J. G. Ceder, and M. L. Weiss, Uniform limits of Darboux functions, Colloq. Math. 15(1) (1966), 65-77. MR 33:5794
  • 3. J. G. Ceder, Differentiable roads for real functions, Fund. Math. 65 (1969), 351-358. MR 40:4398
  • 4. J. G. Ceder and T. L. Pearson, Insertion of open functions, Duke Math. J. 35 (1968), 277-288. MR 36:6556
  • 5. K. Ciesielski and A. W. Miller, Cardinal invariants concerning functions, whose sum is almost continuous, Real Anal. Exchange 20(2) (1994-95), 657-672. MR 96h:26003
  • 6. K. Ciesielski and I. Rec{\l}aw, Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange 21(2) (1995-96), 459-472. MR 97f:26003
  • 7. U. B. Darji and P. D. Humke, Every bounded function is the sum of three almost continuous bounded functions, Real Anal. Exchange 20(1) (1994-95), 367-369. MR 95m:26004
  • 8. H. W. Ellis, Darboux properties and applications to non-absolutely convergent integrals, Canad. J. Math. 3 (1951), 471-485. MR 13:332d
  • 9. R. G. Gibson and F. Roush, Connectivity functions with a perfect road, Real Anal. Exchange 11(1) (1985-86), 260-264.
  • 10. Z. Grande, A. Maliszewski, and T. Natkaniec, Some problems concerning almost continuous functions, Real Anal. Exchange 20(2) (1994-95), 429-432.
  • 11. A. Maliszewski, Sums of bounded Darboux functions, Real Anal. Exchange 20(2) (1994-95), 673-680. MR 96f:26002
  • 12. T. Natkaniec, Almost continuity, Real Anal. Exchange 17(2) (1991-92), 462-520. MR 93e:54009
  • 13. -, Extendability and almost continuity, Real Anal. Exchange 21(1) (1995-96), 349-355. MR 97g:26002
  • 14. T. Radakovi\v{c}, Über Darbouxsche und stetige Funktionen, Monatsh. Math. Phys. 38 (1931), 117-122.

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Additional Information

Krzysztof Ciesielski
Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506–6310

Aleksander Maliszewski
Affiliation: Department of Mathematics, Pedagogical University, Arciszewskiego 22, 76–200 Słupsk, Poland
Email: wspb05@pltumk11.bitnet

Keywords: Peripheral continuity, almost continuity, connectivity, extendability
Received by editor(s): March 28, 1996
Received by editor(s) in revised form: August 11, 1996
Additional Notes: This work was partially supported by NSF Cooperative Research Grant INT-9600548
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society

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