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.
@article{bwmeta1.element.doi-10_2478_BF02475210,
author = {Petr H\'ajek and Michal Johanis},
title = {Smooth approximations without critical points},
journal = {Open Mathematics},
volume = {1},
year = {2003},
pages = {284-291},
zbl = {1033.46037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475210}
}
Petr Hájek; Michal Johanis. Smooth approximations without critical points. Open Mathematics, Tome 1 (2003) pp. 284-291. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475210/
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