The semi-Borel classification of the extreme path derivatives (Q913002)

This paper continues author's previous paper published in Real Anal. Exch. 13 (1987/88), No.2, 373-389 (1989; Zbl 0657.26004) and it deepens and completes some results of the author and others on the Borel classification of \(D_{f,{\mathcal E}}\) and the extreme derivatives \(\sup D_{f,{\mathcal E}}\) and \(\inf D_{f,{\mathcal E}}\) when f is a Borel function. Also, there is being worked with the quadruples \({\mathcal E}=(R,{\mathcal T},E,{\mathcal C})\) called derivation systems or shortly DS, where R is the set of all real numbers, \({\mathcal T}\) is a topology on R, E is a multifunction defined on R with some properties, called a derivation path and \(\emptyset \neq {\mathcal C}\subset 2^ R\) and \(\emptyset \not\in {\mathcal C}.\) The main result is Lemma 2.4 on the connection of the multifunction \(D_{f,{\mathcal E}}\) and the set of all fixed points of some multifunctions. If T is a multifunction defined on R the set \(P=\{x\in R:\) \(x\in T(x)\}\) is the set of all fixed points of T. Let \({\mathcal E}=(R,{\mathcal T},E,{\mathcal C})\) be a DS and f: \(R\to R\) be a function. Then for \(z\in <-\infty,\infty>\) it holds \(z\in D_{f,{\mathcal E}}(x)\) iff for every in \(<-\infty,\infty>\) open set G containing z and for any \(U\in {\mathcal T}\) with \(x\in U\) there exists a \(V\in {\mathcal C}\) such that \(V\subset U\cap (E(x)-\{x\})\) and \(f_ 0(x,y)\in G\) for all \(y\in V\), where \(f_ 0(x,y)=(f(x)-f(y))/(x-y)\) for any \(x,y\in R\), \(x\neq y.\) Lemma 2.4. Let \({\mathcal E}=(R,{\mathcal T},E,{\mathcal C})\) be a DS, f: \(R\to R\) a function and the system \({\mathcal C}\) let have the following property: If \(n\in N\), \(\{A_ 1,...,A_ n\}\subset 2^ R\) and \(\cup \{A_ i:\quad i=1,...,n\}\in {\mathcal C}\) then there exists a \(j\in \{1,...,n\}\) such that \(A_ j\in {\mathcal C}\). Then for all \(a,b\in <-\infty,\infty>\), \(a<b\), it holds \[ D^-_{f,{\mathcal E}}(<a,b>)=\cap \{P({\mathcal C}_{{\mathcal T}}(F_{A(E,f,n,a,b)})\quad):\quad i\in N\}, \] where \[ D^- _{f,{\mathcal E}}(<a,b>)=\{t\in R:\quad D_{f,{\mathcal E}}(t)\cap <a,b>\neq \emptyset \}, \] \[ A(E,f,n,a,b)=Gr(E)\cap f_ 0^{-1}((a-1/n,b+1/n)), \] \[ Gr(E)=\{(x,y)\in R\times R:\quad y\in E(x)\}, \] \[ F_{A(E,f,n,a,b)}(x)=\{y\in R:\quad (x,y)\in A(E,f,n,a,b)\} \] for all \(x\in R\) and \({\mathcal C}_{{\mathcal T}}(F_{A(E,f,n,a,b)})=\{x\in R:\) for any \(U\in {\mathcal T}\) with \(x\in U\) there exists such a \(V\in {\mathcal C}\) that \(V\subset U\cap (F_{A(E,f,n,a,b)}(x)-\{x\}).\) In the paragraphs 3 and 4, the author presents results on the classification of \(D_{f,{\mathcal E}}\), \(\sup D_{f,{\mathcal E}}\) and \(\inf D_{f,{\mathcal E}}\). The final paragraph 5 is devoted to the \({\mathcal E}\)- primitives.