Darboux-like Functions within the Class of Hamel Functions

Abstract

In this paper we present a discussion of the relations of the classes of Darboux-like functions within the classes of Hamel functions and Sierpńiski-Zygmund Hamel functions. We prove that the inclusion relations among Darboux-like classes remain valid in both cases (under the assumption of CH for Sierpiński-Zygmund Hamel functions). In particular, assuming CH we prove the existence of a Sierpiński-Zygmund Hamel function which is connectivity but not almost continuous. In addition, we investigate the cardinal number Add $(F_1,F_2)$ in the case when one of the families $F_1,\; F_2$ is Darboux-like or Sierpiński-Zygmund and the other one is the class of Hamel functions, where Add $(F_1,F_2)$ is defined as the smallest cardinality of a family $F \subseteq \mathbb{R}^{\mathbb{R}}$ for which there is no $g \in F_1$ such that $g + F \subseteq F_2$