MiS Preprint Repository

This is the first in a series of two papers.

The range of the gradient of a differentiable real-valued function with a non-empty and bounded support (i.e., a bump) is investigated. For a smooth bump ${}^{}f$ on ${\mathbb{R}}^2$ it is shown that the interior $\mathrm{int}\mathrm{\nabla }f({\mathbb{R}}^2)$ is connected and dense in $\mathrm{\nabla }f({\mathbb{R}}^2)$. A purely topological characterization of such gradient ranges is however impossible. We give an example of a compact set $K\subset {\mathbb{R}}^2$ that is homeomorphic to the closed unit disk, but such that no affine image of $K$ is the gradient range of a smooth bump on ${\mathbb{R}}^2$. For smooth bumps on ${\mathbb{R}}^n$ we show that the gradient range cannot be separated by a totally disconnected set. The proof relies on a Morse-Sard type result involving irreducible separators of ${\mathbb{R}}^n$. Proofs are carried out for a class of ${\mathcal{C}}^1$ functions containing all those whose first order derivatives are Lipschitz or of bounded variation.

Finally, we present an example of a ${\mathcal{C}}^1$-smooth bump on ${\ell }_2$, which has a gradient range with non-empty and disconnected interior, and a ${\mathcal{C}}^{\mathrm{\infty }}$-smooth weak bump on ${\ell }_2$ with the same property.