Darboux Like Functions. Old Problems and New Results

Abstract

487-496: During the 14th Summer School on Real Functions Theory in Liptovsky Ján, Slovakia in 1996, Richard G. (Jerry) Gibson gave a talk concerning Darboux-like functions. Following his talk we wrote the survey article Darboux-like functions, that was published in the Real Analysis Exchange. In that paper we collected not only known facts on the subject but also presented some open problems. In the last two years a considerable number of papers have been written that solve some of the open questions posed in our survey article. In this short abstract we would like to present those solutions and to list the problems that remain open. This paper is based on the talk given by Tomasz (Tomek) Natkaniec at the 15th Summer School on Real Functions Theory in Liptovsky Ján, Slovakia, in 1998. Most of the new results are contained as a part of the Polish-American project it Set Theoretic Analysis and has been described in K. Ciesielski's survey Set Theoretic Real Analysis. You can find these results on the Set Theoretic Analysis web page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html