Sums of Darboux and continuous functions

Details Sums of Darboux and continuous functions Sums of Darboux and continuous functions

1993-04-27

arxiv.org/abs/math/9304205v1

Mathematics

Logic

Scientific paper

It is shown that 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 by B.~Kirchiem and T.~Natkaniec who have 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.

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