Abstract
In any separable Banach space containing c 0 which admits a C k-smooth bump, every continuous function can be approximated by a C k-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.
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Hájek, P., Johanis, M. Smooth approximations without critical points. centr.eur.j.math. 1, 284–291 (2003). https://doi.org/10.2478/BF02475210
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DOI: https://doi.org/10.2478/BF02475210