TITLE: Non-zero smooth functions vanishing on a Cantor set
QUESTION [6 upvotes]: It is easy to give examples of continuous functions $f:[0,1]\to \mathbb R_+\cup\{0\}$ non-zero but vanishing on a Cantor set (ex: Can Cantor set be the zero set of a continuous function?). It is clearly non-true for analytic functions. My question is: 

Are there uniformly continuous non-zero functions vanishing on a Cantor set?
Are there α-Hölder continuous non-zero functions vanishing on a Cantor set?
Are there continuously differentiable non-zero functions vanishing on a Cantor set?

REPLY [6 votes]: 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).