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:

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?

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:

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?

It is easy to construct by hand 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:

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?

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:

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?