On the path derivatives (Q1110667)

The paper is a continuation of the concept of the path derivative introduced by Bruckner, O'Malley and Thomson in which the system E of paths is regarded, as by Alikhani-Koopaei, as a multifunction. There are studied various properties of primitives and the properties of the multifunction of the E-derived numbers and the extreme path derivatives. The proofs are based on various generalized types of continuity and measurability of the system of paths. The author's concept of the generalized system of paths, of the continuity, of the measurability and generalized path-differentiation is as follows: Let (R,\({\mathcal O})\) be the usual topological space of all reals, \((R^*,{\mathcal O}^*)\) be the usual two-points compactification of \((R,{\mathcal O}),\) \({\mathcal J}\) be a topology on R which is finer than \({\mathcal O}\) and f be a real function of a real variable. The system E is a quadruple \((R,{\mathcal J},E,{\mathcal C}),\) where \(E:R\to 2^ R\) is a multifunction, i.e. \(E(x)\neq \emptyset\) for all \(x\in R\) and \(\emptyset \neq {\mathcal C}\subset 2^ R\) and \(\emptyset \not\in {\mathcal C}.\) Then \(z\in R^*\) is called an E-derived number of f at x iff for any \(G\in {\mathcal O}^*,\) \(z\in G,\) and any \(U\in {\mathcal J},\) \(x\in U,\) there exists an \(A\in {\mathcal C}\) such that \[ A\subset (U\cap E(x)-\{x\})\cap \{y\in R:\frac{f(x)-f(y)}{x-y}\in G\}. \] Let D(f,E,x) be the set of all E- derived numbers of f at x, \(\ker (f,E)=\{x\in R:D(f,E,x)\cap R\neq \emptyset \},\) \(\sup D(f,E,x)\) and \(\inf D(f,E,x)\) are called extreme E- derivatives of f at x, if \(D(f,E,x)\neq \emptyset\) and z is called E- derivative of f at x iff \(D(f,E,x)=\{z\}.\) Let \(C({\mathcal J},f,x)\) be the cluster set of f at x, i.e. \(C({\mathcal J},f,x)=\{y\in R^*:\) for any \(V\in {\mathcal J},\) \(x\in V\) and any \(U\in {\mathcal O}^*,\) \(y\in U,\) there exists an \(A\in {\mathcal C}\) such that \(A\subset f^{-1}(U)\cap V\}.\) f is called \((C,{\mathcal J})\)-continuous at x iff \(f(x)\in C({\mathcal J},f,x)\) and f is \((C,{\mathcal J})\)-continuous at any \(x\in \ker (f,E).\) f is \({\mathcal J}\)- measurable iff \(f^{-1}(G)\) has the \({\mathcal J}\)-Baire property for any \(G\in {\mathcal O}^*.\) The author says that \(z\in R^*\) is an Es-derived number of f at x iff \(x\in D_{{\mathcal J}}(E(x))\) and for any \(G\in {\mathcal O}^*,\) \(z\in G,\) there exist an \(H\in {\mathcal J}\) and a set \[ A\subset (E(x)\cap H-\{x\})\cap \{y\in R:\frac{f(x)-f(y)}{x-y}\in G\} \] such that \(E(x)\cap (H-D_{{\mathcal J}}(A))\) is of the \({\mathcal J}\)-first category, where \(D_{{\mathcal J}}(A)=\{y\in R:\) for any \(V\in {\mathcal J},\) \(y\in V,\) the set \(V\cap A\) is of the \({\mathcal J}\)-second category\(\}\). If \(D(f,Es,x)=\{z\in R^*:\) z is an Es-derived number of f at \(x\}\neq \emptyset\), we put \(\underline f'_{Es}(x)=\inf D(f,Es,x)\) and \(\bar f'_{Es}(x)=\sup D(f,Es,x).\) I mention only two of the author's results. If for any \(x\in R\) the set E(x) is of the \({\mathcal O}\)-second category at x and has the \({\mathcal O}\)-Baire property and \(f:R\to R\) has a finite E- derivative on an \({\mathcal O}\)-dense set, then f is \({\mathcal O}\)-measurable (Theorem 2.11 (a)). Let (R,\({\mathcal J})\) be a Baire space, E(x) be unilateral on \({\mathcal J}\)- residual set T, \(x\in D_{{\mathcal J}}(E(x))\) and E(x) has the \({\mathcal J}\)- Baire property for any \(x\in T.\) Then \(f:R\to R\) is \({\mathcal J}\)-measurable if \(\underline f'_{Es}(x)>-\infty \quad (\bar f'_{Es}(x)<\infty)\) holds except for a set of the \({\mathcal J}\)-first category (Theorem 3.10). This theorem gives the positive answer for a problem of the reviewer. The author's results about extreme generalized path derivatives are new or generalizations of known assertions.