An auto-homeomorphism of a Cantor set with derivative zero everywhere

We construct a closed bounded subset XX of RR with no isolated points which admits a differentiable bijection f:X→Xf:X→X such that f′(x)=0f′(x)=0 for all x∈Xx∈X. We also show that any such function admits a restriction f↾Pf↾P to an uncountable closed P⊆XP⊆X forming a minimal dynamical system. The existence of such a map ffseems   to contradict several well know results. The map ff marks a limit beyond which Banach Fixed-Point Theorem cannot be generalized.