An analog of Sard's theorem for $C^1$-smooth functions of two variables

The main result of the article is
Theorem 1. {\it Let $v\colon\Omega\to\mathbb R$ be a $C^1$-smooth function on a domain $\Omega\subset\mathbb R^2$. Suppose that $0\notin\operatorname{Cl}\operatorname{Int}D_v(\Omega)$. Then the measure of the image of the set of critical points equals zero.}