Abstract
Assume a polynomially bounded o-minimal structure R on the ordered field of real numbers R is given, having the following property: for somen≥2, for anyR-definableC∞-functionf:Rn−1×(0,∞)→Rof n-variablesx1,...,xn, if all the partial derivatives∂αf(α∈Nn)have continuous extensions toRn−1×[0,∞), then f has anR-definableC∞-extension to a neighborhood of 0 inRn.Then every R-definable C∞-function is real analytic.
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