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

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Gaspari T.

Studia Mathematica

2002

tom 153

nr 1

81-99

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

Ograniczanie wyników

1 Studia Mathematica

1 Gaspari T.

1 2002

Wyniki wyszukiwania

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