Abstract
We show that the gradient range of a $\C^1$-smooth function $b$ on the plane is regularly closed (i.e., it is the closure of its interior), provided $b$ has non-empty bounded support and the gradient $\grad b$ admits a modulus of continuity $\omega = \omega (t)$ that satisfies $\omega (t)/\sqrt{t} \to 0$ as $t \searrow 0$. Furthermore, under the same smoothness hypothesis, we show that the gradient range of a function $b \fcolon \Rn \to \R$ with non-empty bounded support has the topological dimension at least two at points of a dense subset.
The proof relies on a new Morse-Sard type result.