Some properties of generalized derivatives (Q791699)

In this short article, the author presents a few of the main ideas that have proved useful in two articles [\textit{A. M. Bruckner, R. J. O'Malley} and the author, Trans. Am. Math. Soc. 283, 97-125 (1984; Zbl 0541.26003); the author, J. Lond. Math. Soc., II. Ser. 27, 43-50 (1983; Zbl 0523.26004)]. The author considers the following scheme for his generalized derivatives. Let \(R=(-\infty,\infty)\) and \(\{S(x):x\in R\}\) be a set of subsets of R having for each \(x\in R\) the properties: \((i)\sigma \in S(x)\) implies \(x\in \sigma, (ii)\quad \{x\}\not\in S(x),\) (iii) if \(\sigma \in S(x)\) and \(\sigma \subset \sigma '\subset R,\) then \(\sigma '\in S(x)\) and (iv) if \(\sigma \in S(x)\) and \(\eta>0\), then \(\sigma \cap(x-\eta,x+\eta)\in S(x).\) If \(f:R\to R\) and \(x\in R,\) then the author defines: \[ \underline D_ Sf(x)=\sup \{\inf \{\frac{f(y)- f(x)}{y-x}:y\in \sigma,y\neq x\}:\sigma \in S(x)\}, \] \(D_ Sf(x)\) is the unique number (if there exists) for which the set \[ \{y\in R:f(y)-f(x)- D_ Sf(x)(y-x)| \leq \eta | y-x| \} \] belongs to S(x) for all positive \(\eta\) and \[ \underline D^ \sim(x)=\lim \inf \{\frac{f(y)-f(z)}{y-z}:(y,z)\to(x,x)\quad and\quad y\neq z\}. \] In the described differentiable scheme many properties of derivatives are based on two elementary types of assumptions: the porosity assumptions and the intersection assumptions. The proof of the generalization of the well- known theorem of W. H. Young for Dini derivatives of a continuous function is based on some porosity assumptions of sets of S(x) : If \(f:R\to R\) is continuous and if for each \(x\in R\) and each \(\sigma \in S(x)\) the set \(\sigma\) has either the right or the left porosity less than 1, then the set \(\{x\in R:\underline D_ Sf(x)\neq \underline D^ \sim(x)\}\) is of the first category in R. The paper ends with the following theorem: ''If \(f:R\to R\) has a derivative \(D_ S\) at each \(x\in R,\) then f is in the first class of Baire whenever \(\{S(x):x\in R\}\) has the following intersection property: There exists a function \(\delta:R\to(0,\infty)\) such that both sets \(\sigma_ x\cap \sigma_ y\cap(-\infty,x)\) and \(\sigma_ x\cap \sigma_ y\cap(y,\infty)\) are non empty for each \(x,y\in R\) and each \(\sigma_ x\in S(x)\) and each \(\sigma_ y\in S(y)\) for which \(0<y-x<\min(\delta(x),\delta(y)).''\) The mentioned intersection property is in his proof essential.