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Smooth approximations without critical points

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Hájek P., Johanis M.

Open Mathematics

2003

tom 1

nr 3

284-291

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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1 Open Mathematics

1 Hájek P.

1 Johanis M.

1 2003

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Smooth approximations without critical points