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• # Artykuł - szczegóły

## Studia Mathematica

2002 | 153 | 1 | 81-99

## On the range of the derivative of a real-valued function with bounded support

EN

### Abstrakty

EN
We study the set f'(X) = {f'(x): x ∈ X} when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a $C^{p}$-smooth bump b:X → ℝ such that $||b^{(p)}||_{∞}$ is finite, then any connected open subset of X* containing 0 is the range of the derivative of a $C^{p}$-smooth bump. We also study the finite-dimensional case which is quite different. Finally, we show that in infinite-dimensional separable smooth Banach spaces, every analytic subset of X* which satisfies a natural linkage condition is the range of the derivative of a C¹-smooth bump. We then find an analogue of this condition in the finite-dimensional case

81-99

wydano
2002

### Twórcy

autor
• Mathématiques Pures de Bordeaux (MPB), UMR 5467 CNRS, Université Bordeaux 1, 351, cours de la Libération, 33400 Talence, France