Two Examples Concerning Extendable and Almost Continuous Functions

Authors

Krzysztof Ciesielski, West Virginia UniversityFollow

Document Type

Article

Publication Date

1999

College/Unit

Eberly College of Arts and Sciences

Department/Program/Center

Mathematics

Abstract

The main purpose of this paper is to describe two examples. The first is that of an almost continuous, Baire class two, non-extendable function f:[0,1]-->[0,1] with a G_\delta graph. This answers a question of Gibson. The second example is that of a connectivity function F:R²-->R with dense graph such that F^-1(0) is contained in a countable union of straight lines. This easily implies the existence of an extendable function f:R-->R with dense graph such that f^-1(0) is countable.

We also give a sufficient condition for a Darboux function f:[0,1]-->[0,1] with a G_\delta graph whose closure is bilaterally dense in itself to be quasicontinuous and extendable.

Digital Commons Citation

Ciesielski, Krzysztof, "Two Examples Concerning Extendable and Almost Continuous Functions" (1999). Faculty & Staff Scholarship. 833.
https://researchrepository.wvu.edu/faculty_publications/833