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.**