Gradient ranges of bumps on the plane

Abstract:

For a $\mathcal {C}^1$-smooth bump function $b \colon {\mathbb R}^{2} \to \mathbb {R}$ we show that the gradient range $\nabla b( {\mathbb R}^{2} )$ is the closure of its interior, provided that $\nabla b$ admits a modulus of continuity $\omega = \omega (t)$ satisfying $\omega (t)/\sqrt {t} \to 0$ as $t \searrow 0$. The result is a consequence of a more general result about gradient ranges of bump functions $b \colon {\mathbb R}^{n} \to \mathbb {R}$ of the same degree of smoothness. For such bump functions we show that for open sets $G \subset {\mathbb R}^{n}$, either the intersection $\nabla b( {\mathbb R}^{n}) \cap G$ is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension $n$ of the space.