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Consider an o-minimal expansion of the real field. We show that definable Lipschitz continuous maps can be definably fine approximated by definable continuously differentiable Lipschitz maps whose Lipschitz constant is close to that of the original map.

We study the Hardy field associated with an o-minimal expansion of the real numbers. If the set of analytic germs is dense in the Hardy field, then we can definably analytic separate sets in R 2 , and we can definably analytic approximate definable continuous unary functions.

Let R be an o-minimal expansion of a real closed field. We show that the definable infinitely Peano differentiable functions are smooth if and only if R is polynomially bounded.

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