On Some Subclasses of Baire 1 Functions

Abstract

We will consider the classes of first return continuous, weakly first return continuous with respect to some trajectory and almost continuous functions in the sense of Stallings and we will show that for the Baire $1$ functions the classes of functions mentioned above are equal. Moreover we will define the class $\mathcal{B}_{1}^{\mathcal{F}}$ of strongly $\mathcal{F}$-almost everywhere first return recoverable functions and shall present the relation between family $\mathcal{B}_{1}^{\mathcal{F}}$ and Baire $1$ functions.