Let $d\in\mathbb N$, $\Omega$ be a $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, $f:\partial\Omega\to\mathbb R$ be $C^1$-differentiable, $O\subseteq\mathbb R^d$ be open with $\partial\Omega\subseteq O$ and $$\tilde f(x):=f(\operatorname{cl}(x.\partial\Omega))\;\;\;\text{for }x\in O,$$ where $\operatorname{cl}(x.\partial\Omega)$ is the closet point in $\partial M$ to $x\in O$.
How can we show that $\tilde f\in C^1(O)$ and $$\langle\nabla\tilde f(x),\nu_{\partial\Omega}(x)\rangle=0\;\;\;\text{for all }x\in\partial\Omega,\tag1$$ where $\nu_{\partial\Omega}$ denotes the unique outward pointing unit normal field on $\partial\Omega$.
I'm not sure, but maybe we need to restrict $\tilde f$ to $$T_\varepsilon:=\{x\in O:\operatorname{dist}(x,\partial\Omega)<\varepsilon\}$$ for some suitable $\varepsilon>0$.