I'm eager to prove the following proposition, and there remains one difficulty I need to conquer. Please do me a favor. Thank you very much.
Proposition Let $U$ be a bounded open set in $\mathbb{R}^n$ and let $u:\overline{U}\to\mathbb{R}^n$ be $C^1$. Suppose $x_0\in\mathbb{R}^n\setminus u(\partial U)$ is a regular value of $u$. If $\eta:\mathbb{R}^n\to\mathbb{R}$ is $C^1$ with $\int_{\mathbb{R}^n}\eta(z)dz=1$ and $\mathrm{supp}(\eta)\subseteq B(x_0,r)$, then for $r$ small enough, $$\int_U\eta(u(x))\det(Du(x))dx\in\mathbb{Z}.$$
Here $B(x_0,r)$ stands for an open ball of center $x_0$ and radius $r$, and $\det(Du(x))$ denotes the Jacobian determinant of $u$ at $x$.
Attempt It's not difficult to show that $u^{-1}(\{x_0\})$ is finite, and I shall not discuss the case that $u^{-1}(\{x_0\})$ is trivially empty. Let $u^{-1}(\{x_0\})=\{a_1,\ldots,a_k\}$. It's not too hard to find $\epsilon>0$ s.t.
(i) $B(a_i,\epsilon)\subseteq U$ for all $i$.
(ii) $B(a_i,\epsilon)\cap B(a_j,\epsilon)=\emptyset$ if $i\not= j$.
(iii) For each $i$, either $\det(Du)>0$ in $B(a_i,\epsilon)$, or $\det(Du)<0$ in $B(a_i,\epsilon)$.
Note that one may employ continuity of the Jacobian determinant to accomplish (iii). Now use the inverse function theorem to choose $r_1>0$ s.t. $B(x_0,r_1)$ is contained in each $u(B(a_i,\epsilon))$. Here comes the problem. I don't know how to find $r_2>0$ s.t. $$u^{-1}(B(x_0,r_2))\subseteq\bigcup_{i=1}^k B(a_i,\epsilon).$$ If such $r_2$ does exist, then by choosing $0<r\leq\min\{r_1,r_2\}$, I can prove that the integral in question is equal to $$\sum_{i=1}^k\mathrm{sgn}(\det(Du(a_i))),$$ which is without doubt an integer. Please give me a hand. Thank you.