Differentiability almost everywhere of functions of several variables

Abstract

1. The purpose of this paper is to extend the results of C. J. Neugebauer in [3], where necessary and sufficient conditions for a measurable function of a real variable to be equivalent to one which is differentiable are given, to the case of higher order differentiation and to functions of several real variables. Letf: En-*R be a measurable function from n-dimensional Euclidean space to the reals. Q will denote a cube in En; and if E is a measurable set in En, we denote its measure by I EJ . All functions and sets will be assumed to be measurable. Suppose that we can writef(xo+t) -Pz0(t) +Rx0(t) where P,0(t) is a polynomial of degree < k 1, then we say f has a k-I derivative at x0 in the LP sense (1 < p? oo ) if (1) {-f |~~ vRxo(t) JPdt} = o(p ), p o0