We prove that if X is an infinite-dimensional Banach space with $C^p$ smooth partitions of unity then X and X∖ K are $C^p$ diffeomorphic for every weakly compact set K ⊂ X.

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 $\left|\right|b^{\left(p\right)}{\left|\right|}_{\infty }$ 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 survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of a mapping F...

A subset of ${ℝ}^d$ is called a universal differentiability set if it contains a point of differentiability of every Lipschitz function $f:{ℝ}^d\to ℝ$. We show that any universal differentiability set contains a ‘kernel’ in which the points of differentiability of each Lipschitz function are dense. We further prove that no universal differentiability set may be decomposed as a countable union of relatively closed, non-universal differentiability sets.