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).

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).