Smooth approximations without critical points

✍ Scribed by Petr Hájek; Michal Johanis

• Book ID: 111487495
• Publisher: SP Versita
• Year: 2003
• Tongue: English
• Weight: 229 KB
• Volume: 1
• Category: Article
• ISSN: 1895-1074
• DOI: 10.2478/bf02475210

✦ Synopsis

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 rst 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.