Properties of the $C^1$-smooth functions with nowhere dense gradient range

One of the main results of the present article is as follows
Theorem. {\it Let $v\colon\Omega\to\mathbb R$ be a $C^1$-smooth function on a domain $\Omega\subset\mathbb R^2$. Suppose that $\operatorname{Int}\nabla v(\Omega)=\varnothing$. Then, for every point $z\in\Omega$, there is a straight line $L\ni z$ such that $\nabla v\equiv\mathrm{const}$ on the connected component of the set $L\cap\Omega$ containing $z$}.
Also, we prove that, under the conditions of the theorem, the range of the gradient $\nabla v(\Omega)$ is locally a curve and this curve has tangents in the weak sense and the direction of these tangents is a function of bounded variation.