On the theorem of Rademacher (Q1194681)

Suppose that \(X\) is a compact (e.g., of Lebesgue measure zero) subset of \(\mathbb{R}^ N\) and \(f: X\to\mathbb{R}^ m\) is locally Lipschitz. Does it follow that \(f\) has to be (strongly) differentiable at some point \(x\in X\)? For the case of sets of measure zero on the line and \(m=1\) the problem is solved in the negative. In \(\mathbb{R}^ N\), \(N\geq 2\), there is a set of measure zero such that every real-valued locally Lipschitz function has points of differentiability in it [\textit{D. Preiss}, J. Funct. Anal. 91, No. 2, 312-345 (1990; Zbl 0711.46036)]. In the present paper, the author constructs a large family of Cantor subsets \(X\) of \(\mathbb{R}^ N\) and bi- Lipschitz maps \(f: X\to f(X)\subset H\), where \(H\) is an infinite- dimensional Hilbert space, such that \(f\) is not strongly differentiable at any point of \(X\). (By a strong derivative \(D\) of \(f\) at \(x\in X\) it is meant a linear map \(D: \mathbb{R}^ N\to H\) with the property that for any \(\varepsilon>0\) there is a \(\delta>0\) such that \(\| f(y)-f(x)-D(y- x)\|<\varepsilon\| y-x\|\) provided that \(x,y\in X\) and \(\| y- x\|<\delta\).) Furthermore, the image \(M=f(X)\subset H\) has the property that for any differentiable map \(F: [0,1]^ N\to H\) with \(dF(x)\) nonsingular for all \(x\in [0,1]^ N\) the set \(F^{-1}(M)\) is a finite set. Hence, such a map \(f\) can agree with a nonsingular differentiable map on at most a finite set.