Path derived numbers and path derivatives of continuous functions with respect to continuous systems of paths (Q1766450)

A system of paths \(R= \{R_x: x\in [0,1]\}\) is called a continuous system of paths if the function \(P: x\to R_x\) is continuous, when \(R\) is equipped with the Hausdorff metric (all paths are supposed to be compact). The main result of the paper is Theorem 5, which says that if \(f\) is a continuous function and both extreme derivatives of \(f\) with respect to a continuous system of paths are finite, then there exists a dense open set on which \(f\) is differentiable for almost all \(x\) belonging to this set. The paper contains also the comparison of the behaviour of first return derivatives and path derivatives coming from a continuous system of paths.