Baire one path systems, Baire^* one path systems and path derivatives

In this paper, we study the class of Baire one path systems as a metric space and as a multifunction and show that some known classes of path systems such as continuous path systems, composite continuous path systems, and first return path systems are subclasses of the class of Baire one path systems. In particular, we show that a path system is composite continuous if and only if it is Baire^* one path system. We also show that for a composite continuous path system E = {Ex : x ∈ [0, 1]}, the upper (respectively, lower) extreme path derivatives f'_E(respectively, f'_E) of a function f ∈ B1 are upper (respectively, lower) Baire two functions, and when f is E-differentiable, f'_E ∈ B2.