On absolute Peano derivatives
In a recent survey [Real Anal. Exch. 7, 5-23 (1982; preceding review)], \textit{M. J. Evans} and \textit{C. E. Weil} asked if the absolute Peano derivative of \textit{M. Laczkovich} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 21, 83-97 (1978; Zbl 0425.26005)] has the -M,M and Z properties, properties enjoyed by most other derivatives. The present author has defined generalized Peano derivatives in Trans. Am. Math. Soc. 275, 381-396 (1983; Zbl 0506.26006). A function f is said to have generalized n-th Peano derivative at t if f is continuous in some neighbourhood of t and if for some positive integer k the k-th primitive of f has a \((k+n)\)-th Peano derivative at t. In that paper the author has shown also that every absolute Peano derivative is a generalized Peano derivative. In the present article the author proves that generalized Peano derivatives have both of the above properties, thus answers the question of Evans and Weil positively. His proof uses basic results from his article quoted above, certain ideas of Mařík, O'Malley and Weil, and the extension to generalized Peano derivatives of the work of Sargent on Peano derivatives.
scientific article; zbMATH DE number 3849527