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?][1]). It is clearly non-true for analytic functions. My question is: 

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

  [1]: https://mathoverflow.net/questions/24034/can-cantor-set-be-the-zero-set-of-a-continuous-function