Sums of Darboux and continuous functions

Juris Steprans

Fundamenta Mathematicae (1995)

  • Volume: 146, Issue: 2, page 107-120
  • ISSN: 0016-2736

Abstract

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It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent - the model used is iterated Sacks forcing - that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of set theory there are universally bad Darboux functions, Darboux functions whose sum with any nowhere constant, continuous function fails to be Darboux.

How to cite

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Steprans, Juris. "Sums of Darboux and continuous functions." Fundamenta Mathematicae 146.2 (1995): 107-120. <http://eudml.org/doc/212055>.

@article{Steprans1995,
abstract = {It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent - the model used is iterated Sacks forcing - that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of set theory there are universally bad Darboux functions, Darboux functions whose sum with any nowhere constant, continuous function fails to be Darboux.},
author = {Steprans, Juris},
journal = {Fundamenta Mathematicae},
keywords = {sums of Darboux functions and continuous functions; iterated Sacks forcing},
language = {eng},
number = {2},
pages = {107-120},
title = {Sums of Darboux and continuous functions},
url = {http://eudml.org/doc/212055},
volume = {146},
year = {1995},
}

TY - JOUR
AU - Steprans, Juris
TI - Sums of Darboux and continuous functions
JO - Fundamenta Mathematicae
PY - 1995
VL - 146
IS - 2
SP - 107
EP - 120
AB - It is shown that for every Darboux function F there is a non-constant continuous function f such that F + f is still Darboux. It is shown to be consistent - the model used is iterated Sacks forcing - that for every Darboux function F there is a nowhere constant continuous function f such that F + f is still Darboux. This answers questions raised in [5] where it is shown that in various models of set theory there are universally bad Darboux functions, Darboux functions whose sum with any nowhere constant, continuous function fails to be Darboux.
LA - eng
KW - sums of Darboux functions and continuous functions; iterated Sacks forcing
UR - http://eudml.org/doc/212055
ER -

References

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  1. [1] J. E. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271-288. Zbl0427.03043
  2. [2] A. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, Berlin, 1978. Zbl0382.26002
  3. [3] A. Bruckner and J. Ceder, Darboux continuity, Jahresber. Deutsch. Math.-Verein. 67 (1965), 100. Zbl0144.30003
  4. [4] A. Bruckner and J. Ceder, On the sums of Darboux functions, Proc. Amer. Math. Soc. 51 (1975), 97-102. Zbl0306.26002
  5. [5] B. Kirchheim and T. Natkaniec, On universally bad Darboux functions, Real Anal. Exchange 16 (1990-91), 481-486. Zbl0742.26010
  6. [6] P. Komjáth, A note on Darboux functions, ibid. 18 (1992-93), 249-252. 
  7. [7] A. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575-584. Zbl0527.03031
  8. [8] T. Radakovič, Über Darbouxsche und stetige Funktionen, Monatsh. Math. Phys. 38 (1931), 117-122. Zbl57.0299.02
  9. [9] S. Saks, Theory of the Integral, Hafner, New York, 1937. 

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