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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
##### Keywords:
Rolle theorem; smooth norm; Brouwer fixed point theorem; bump
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