Path derived numbers and path derivatives of continuous functions (Q2496984)

A path at \(x\in [0,1]\) is a set \(E_x\subset [0,1]\) such that \(x\in E_x\) and \(x\) is a point of accumulation of \(E_x\). A system of paths is a mapping \([0,1]\ni x \overset{E}{\mapsto} E_x\) where each \(E_x\) is a path at \(x\). The author introduces the following definition: Let \(E\) be a system of paths with each \(E_x\) compact. \(E\) is called \(\sigma\)-continuous if there exists a sequence of closed sets \(A_i\) such that \([0,1]=\bigcup_{i=1}^\infty A_i\) and \(E\) is continuous on \(A_i\) for each \(i\). The following theorems are announced in the paper. Theorem 1. If \(F:[0,1]\to \mathbb{R}\) is compositely differentiable to one of its bilateral derivative functions \(f\), then \(F\) is path differentiable to \(f\) with respect to a \(\sigma\)-continuous system of paths \(E\). Theorem 2. Let \(E\) be a \(\sigma\)-continuous system of paths and \(f:[0,1]\to \mathbb(R)\) be a continuous function. If \(f\) is \(E\)-differentiable, then the set \(\{x:f_E'\) is continuous\(\}\) is dense in \([0,1]\). Theorem 3. Let \(f:[0,1]\to \mathbb{R}\) be a continuous function and let \(E\) be a \(\sigma\)-continuous system of paths, then the extreme path derivatives \(f_E'\) and \(\bar{f}_E'\) are elements of \(B_2\).