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Preprint Number 627

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627. A. Thamrongthanyalak
Definable smoothing of continuous functions

Submission date: 22 September 2013.


Let R be an o-minimal expansion of a real closed field. Given definable continuous functions f : U → R and ε : U → (0, +∞), where U is an open subset of R^n, we construct a definable C^m-function g: U → R with |g(x)-f(x)| < ε(x) for all x ∈ U. Moreover, we show that if f is uniformly continuous, then g can also chosen to be uniformly continuous.

Mathematics Subject Classification: 03C64

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