On the range of the derivative of a real-valued function with bounded support
Volume 153 / 2002
Abstract
We study the set $f'(X)=\{f'(x): x \in X\}$ when $f:X\rightarrow \mathbb R $ is a differentiable bump. We first prove that for any $C^2$-smooth bump $f: \mathbb R^2 \rightarrow \mathbb 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\rightarrow \mathbb R $ such that $\| b ^{(p)} \| _{\infty} $ is finite, then any connected open subset of $X^{\ast}$ 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 $X^{\ast}$ which satisfies a natural linkage condition is the range of the derivative of a $C^1$-smooth bump. We then find an analogue of this condition in the finite-dimensional case
