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The failure of Rolle’s theorem in infinite-dimensional Banach spaces. (English) Zbl 0995.46025
A bump is nonzero real function with bounded support. The authors prove that if a Banach space \(X\) admits a \(C^p\) smooth (Lipschitz) bump then it admits another \(C^p\) smooth (Lipschitz) bump \(f:X\to [0,1]\) with the property that \(f'(x)\neq 0\) for all x in the interior of the support of \(f\). This is applied to discussing Rolle’s theorem, deleting diffeomorphisms, and Brouwer fixed points in infinite dimensions.

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
47H10 Fixed-point theorems
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