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

T. Gaspari. "On the range of the derivative of a real-valued function with bounded support." Studia Mathematica 153.1 (2002): 81-99. <http://eudml.org/doc/284401>.

@article{T2002,
abstract = {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},
author = {T. Gaspari},
journal = {Studia Mathematica},
keywords = {Fréchet differentiable; -smooth bump; range of the derivative; infinite-dimensional separable Banach space},
language = {eng},
number = {1},
pages = {81-99},
title = {On the range of the derivative of a real-valued function with bounded support},
url = {http://eudml.org/doc/284401},
volume = {153},
year = {2002},
}

TY - JOUR
AU - T. Gaspari
TI - On the range of the derivative of a real-valued function with bounded support
JO - Studia Mathematica
PY - 2002
VL - 153
IS - 1
SP - 81
EP - 99
AB - 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
LA - eng
KW - Fréchet differentiable; -smooth bump; range of the derivative; infinite-dimensional separable Banach space
UR - http://eudml.org/doc/284401
ER -