ON THE THEOREM OF RADEMACHER

Abstract

It is shown that there exist Cantor subsets $X$ of ${ℝ}^N$ and bi-Lipschitz maps $f:X\to f(X)\subset \mathcal{H}$, where $\mathcal{H}$ is an infinite dimensional Hilbert space, such that $f$ is not strongly differentiable at any point of $X$. Furthermore, for each such $X$ and $f$ the image $M=f(X)$ has the property that for any $N\ge 1$ and any differentiable map $F:{[0,1]}^N\to \mathcal{H}$ with $dF(x)$ nonsingular for all $x\in {[0,1]}^N$, the set $F^{-1}(M)$ is a finite set. Hence, $f$ can agree with a nonsingular differentiable map at most on a finite set.