Condition $\mathcal{B}$ and Baire $1$ generalized derivatives

Ordered pairs (F, f) of real-valued functions on [0,1] which satisfy the condition that every perfect set M contains a dense ${G_\delta }$ set K such that $F\backslash M$ is differentiable to f on K are shown to play a key role in several types of generalized differentiation. In particular, this condition is utilized to prove the equivalence of selective differentiation and various forms of path differentiation under the assumption that the derivatives involved are of Baire class 1, thereby providing an affirmative answer, for Baire 1 selective derivatives, to a question raised in [Trans. Amer. Math. Soc 283 (1984), 97-125].