If $Y\subset X$ are open sets in $\mathbb{R}^n$,then $Y$ is said to have a $C^1$ boundary in $X$ if for every boundary point $x_{0}\in X$ of $Y$ one can find a $C^1$ function $\rho$ in a neighborhood $X_{0}$ of $x_{0}$ such that $ \rho(x_0)=0,\quad d\rho(x_{0})\ne 0,\quad Y\cap X=\{x\in X_{0};\rho(x)<0\} $ Then my question is how to prove that there exists a function$\rho\in C^{1}(X)$ such that $\rho=0$,$d\rho\ne 0$ on the boundary $\partial Y$ in $X$ and $ Y\cap X=\{x\in X;\rho(x)<0\}$
It's very natural to think about using partition of unity to put the local result global,but I don't know how to make these properties still hold when choosing the partition of unity,and the boundary may be unbound and quite arbitrary.