Mathematics Commons^™

Two Examples Concerning Extendable And Almost Continuous Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

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 …

Compositions Of Darboux And Connectivity Functions, Krzysztof Ciesielski

Faculty & Staff Scholarship

We describe here an example of a Darboux function k from the unit interval I = [0, 1] onto itself such that k is not the composition of any finite collection of connectivity functions from I into I. This answers a question of Ceder [2].

Uniformly Antisymmetric Function With Bounded Range, Krzysztof Ciesielski

Faculty & Staff Scholarship

The goal of this note is to construct a uniformly antisymmetric function f : R → R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson [6]. (See also the list of problems in Thomson [9] and Problem 2(b) from Ciesielski’s survey [5].) A problem of existence of uniformly antisymmetric function f : R → R with finite range remains open.