Multidimensional Riemann derivatives

J. Marshall Ash ¹   ; Stefan Catoiu ²

¹ Department of Mathematics DePaul University Chicago, IL 60614, U.S.A.
² Department of Mathematics DePaul University Chicago, IL 60614

Studia Mathematica, Tome 235 (2016) no. 1, pp. 87-100

Résumé

The well-known concepts of $n$th Peano, Lipschitz, Riemann, Riemann Lipschitz, symmetric Riemann, and symmetric Riemann Lipschitz derivatives of real functions of a single variable have natural extensions to functions of several variables. We show that if a function has any of these $n$th derivatives at each point of a measurable subset of $\mathbb {R}^{d}$ then it has all these derivatives at almost every point of that subset.

Détail

DOI : 10.4064/sm8578-7-2016

Keywords: well known concepts nth peano lipschitz riemann riemann lipschitz symmetric riemann symmetric riemann lipschitz derivatives real functions single variable have natural extensions functions several variables function has these nth derivatives each point measurable subset mathbb has these derivatives almost every point subset