On the averages of Darboux functions

Abstract

Let $\mathbf {A}$ be the family of functions which can be written as the average of two comparable Darboux functions. In 1974 A. M. Bruckner, J. G. Ceder, and T. L. Pearson characterized the family $\mathbf {A}$ and showed that if $\alpha \ge 2$, then $\mathbf {A} \cap \mathbf {B}_\alpha$ is the family of the averages of comparable Darboux functions in Baire class $\alpha$. They also asked whether the latter result holds true also for $\alpha =1$. The main goal of this paper is to answer this question in the negative and to characterize the family of the averages of comparable Darboux Baire one functions.