On differentiability of Peano type functions. III

Abstract:

We show that for all positive natural numbers $m$,$n$ the following two sentences are equivalent: (i) ${2^{{\aleph _0}}} \leqslant {\aleph _n}$; (ii) there exists an onto function $f:{R^n} \to {R^{n + m}}$ ($R$ the set of real numbers) such that at each point of ${R^n}$ at least $n$ coordinates of $f$ are differentiable.