We construct a closed bounded subset X of R with no isolated points which admits a differentiable bijection f from X to X such that f'(x)=0 for all x in X. We also show that any such function admits a restriction to an uncountable closed subset P of X forming a minimal dynamical system. The existence of such a map f seems to contradict several well know results. The map f marks a limit beyond which the Banach Fixed-Point Theorem cannot be generalized.
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Last modified May 25, 2016.