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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 1, Issue 3


Volume 13 (2015)

Smooth approximations without critical points

Petr Hájek / Michal Johanis
Published Online: 2003-09-01 | DOI: https://doi.org/10.2478/BF02475210


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.

Keywords: derivatives; approximation; critical points

Keywords: 46B20; 46G05

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About the article

Published Online: 2003-09-01

Published in Print: 2003-09-01

Citation Information: Open Mathematics, Volume 1, Issue 3, Pages 284–291, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/BF02475210.

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© 2003 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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