Sums of Darboux and continuous functions
Steprans, Juris
Fundamenta Mathematicae, Tome 146 (1995), p. 107-120 / Harvested from The Polish Digital Mathematics Library
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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.

Publié le : 1995-01-01
EUDML-ID : urn:eudml:doc:212055 
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Steprans, Juris. Sums of Darboux and continuous functions. Fundamenta Mathematicae, Tome 146 (1995) pp. 107-120. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv146i2p107bwm/

[00000] [1] J. E. Baumgartner and R. Laver, Iterated perfect set forcing, Ann. Math. Logic 17 (1979), 271-288. | Zbl 0427.03043

[00001] [2] A. Bruckner, Differentiation of Real Functions, Lecture Notes in Math. 659, Springer, Berlin, 1978. | Zbl 0382.26002

[00002] [3] A. Bruckner and J. Ceder, Darboux continuity, Jahresber. Deutsch. Math.-Verein. 67 (1965), 100. | Zbl 0144.30003

[00003] [4] A. Bruckner and J. Ceder, On the sums of Darboux functions, Proc. Amer. Math. Soc. 51 (1975), 97-102. | Zbl 0306.26002

[00004] [5] B. Kirchheim and T. Natkaniec, On universally bad Darboux functions, Real Anal. Exchange 16 (1990-91), 481-486. | Zbl 0742.26010

[00005] [6] P. Komjáth, A note on Darboux functions, ibid. 18 (1992-93), 249-252.

[00006] [7] A. Miller, Mapping a set of reals onto the reals, J. Symbolic Logic 48 (1983), 575-584. | Zbl 0527.03031

[00007] [8] T. Radakovič, Über Darbouxsche und stetige Funktionen, Monatsh. Math. Phys. 38 (1931), 117-122. | Zbl 57.0299.02

[00008] [9] S. Saks, Theory of the Integral, Hafner, New York, 1937.