A. M. Bruckner, R. J. O'Malley, and B. S. Thomson introduced path differentiation as a vehicle for unifying the theory of numerous types of generalized differentiation of real valued functions of a real variable. Part of their classification scheme was based on intersection properties of the underlying path systems. Here, additional light is shed on the relationships between these various types of path differentiation and it is shown how composite differentiation and first return differentiation fit in to this scheme.
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A new theorem in the theory of first return representations of Baire class one functions of a real variable is presented which has as immediate consequences several known characterizations of standard subclasses of the Baire one functions. Further, this theorem yields new insights into how finely Baire one functions can be recovered and yields a characterization of another subclass of Baire one functions.