Given a $W^{1,\infty}$ function and a convex set , how can we find a Lipschitz function?

Question

Let $\Omega \subset \mathbb R^n$ be open, connected and $\Omega=\Omega_1\cup \Omega_2$, where $\Omega_1\cap \Omega_2=\phi, \mu(\Omega_1)>0, \mu(\Omega_2)>0.$ Also we have a function $\theta \in W^{1,\infty}(\Omega;\mathbb R)$ such that $D\theta (x)=\chi_{\Omega_1}(x)p$ for all $x\in \Omega $ a.e. where $p \in S^{n-1}.$

The question is : Let $E\subseteq \Omega$ be a convex set. Then $\exists$ a Lipschitz function $f_E: \mathbb R \to \mathbb R$ such that

1. $\theta(x)=f_E(x.p)$ for all $x\in E,$
2. $f_E^{'}(t)\in \{0,1\}$ for all $t\in \mathbb R$ a.e.

I just need some hints. how to start this one.

I did for $n=1$ taking $\Omega=(0,1)=(0,1]\cup(1,2)$ and it's easy .

Any help is appreciated. Thank you.