On differentiability of Peano type functions

We show that for all positive natural numbers m, n the following two sentences are equivalent: (i) 2NO 1 we present, using notions of classical mathematical analysis, a sentence (Pn equivalent to the sentence 2to 1 the theory ZFC + 2NO = Nn is consistent (whenever the theory ZFC is consistent). Solovay's result implies as well that the theory ZFC + 2No > N, is consistent (w denotes here the order type of the set of all natural numbers). The sentence 2No = N1 is called the Continuum Hypothesis, and its independence of ZFC results from the work of K. G6del and P. J. Cohen. The sentence (On proclaims that there exists an onto function f: Rn __ Rn+m (R denotes the set of real numbers), where m is any natural number > 1, such that at each point of Rn at least n coordinates of f are differentiable. We also analyse the question concerning regularity of such functions. This paper is a continuation of [3 and 4]. Our results generalize theorems from those papers in which the existence of onto functions f:R -* R , where at each point of R at least one coordinate of f is differentiable, is analysed. 1. Auxiliary theorems. Let R denote the set of real numbers and N the set of natural numbers, where i, j, k, 1, m, n, p, r E N. For any sets A and B the set of all functions from A to B is denoted by BA. We treat the cartesian product An as the set A{'.n}. This convention enables us to consider k-dimensional sections of subsets of Rn more easily. The cardinality of a set X is denoted by IXI. Let P = [{1 .. ., m}]n be the set of all subsets of {1, . . ., m}, the cardinality of which is n. For a fixed product Xm and A C Xm, A c P7, a E Xm (or a E XA), let (A)a,A -{x E A: x(i) = a(i) for each i E A}. A family {EA: A E Pn7} is called the (m, n)-Sierpin'ski partition of a set Xm if U{EA: A c P7 } = Xn and, for every A E PT and a E Xm, we have I(EA)a,AI 0. A general theorem about the existence of (Tn,n)Received by the editors April 25, 1983. 1980 Mathematics Subject Classification. Primary 04A30, 26B05.