This answer is to provide a precise statement for a result mentioned by Peter Michor.

> **Theorem (Calderón-Zygmund).**  If $F\subset\mathbb{R}^n$ is closed, then there is a function $f$ such that
> 
>  - $c_1(n)d(x,F)\leq f(x)\leq c_2(n)d(x,F)$, for all $x\in\mathbb{R}^n$,
>  - $f\in C^\infty(\mathbb{R}^n\setminus F)$ and  $$ \left|\frac{\partial^\alpha f}{\partial x^\alpha}(x)\right|\leq
 c_3(n,\alpha) d(x,F)^{1-|\alpha|} \quad \text{for all multiindices
 $\alpha$.} $$

This result says that there is a smooth regularization of a distance function. Originally it was proved in:  

**A.P.Calderón, A.Zygmund,** *A. Local properties of solutions of elliptic partial differential equations.* Studia Math. 20 1961 171–225. 

You can also find this result in  

**E.M. Stein,** *Singular integrals and differentiability properties of functions.* Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 (Theorem 2 in Chapter VI.2).