Regular closed gradient ranges and asymptotic gradient values

We prove that the gradient range of every nonzero compactly supported $C^1$ function on ${\mathbb{R}}^n$ is the closure of its interior, for every $n\ge 1$. More generally, for an arbitrary $C^1$ function, the closure of its gradient range is the union of the closure of the range’s interior and the set of asymptotic gradient values. We also prove that compact smooth Lagrangian submanifolds without boundary in ${\mathbb{R}}^{2n}$ and their images under symplectic homeomorphisms have regular closed momentum projections.