# Smooth approximations without critical points

Open Mathematics (2003)

- Volume: 1, Issue: 3, page 284-291
- ISSN: 2391-5455

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topPetr Hájek, and Michal Johanis. "Smooth approximations without critical points." Open Mathematics 1.3 (2003): 284-291. <http://eudml.org/doc/268737>.

@article{PetrHájek2003,

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.},

author = {Petr Hájek, Michal Johanis},

journal = {Open Mathematics},

keywords = {46B20; 46G05},

language = {eng},

number = {3},

pages = {284-291},

title = {Smooth approximations without critical points},

url = {http://eudml.org/doc/268737},

volume = {1},

year = {2003},

}

TY - JOUR

AU - Petr Hájek

AU - Michal Johanis

TI - Smooth approximations without critical points

JO - Open Mathematics

PY - 2003

VL - 1

IS - 3

SP - 284

EP - 291

AB - 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.

LA - eng

KW - 46B20; 46G05

UR - http://eudml.org/doc/268737

ER -

## References

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