In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted $Conn^*$) contained between the families (widely described in literature) of Darboux Baire 1 functions (${\rm DB}_1$) and connectivity functions ($Conn$). The solutions to our problems are based, among other, on the suitable construction of the ring, which turned out to be in some senses an “optimal construction“. These considerations concern mainly real functions defined on $[0,1]$ but in the last chapter we also extend them to the case of real valued iteratively $H$-connected functions defined on topological spaces.