For a fixed Cantor set $K\subset [0,1]$ (ex: middle third Cantor set) and a function $g:[0,1]\to \mathbb R.$ Is it always possible to find a $C^{\infty}$ map $f:[0,1]\to \mathbb R$ such that $g$ and $f$ coincide in $K?$ 

The case $g=0$ (the constant function $0$) is covered in [Non-zero smooth functions vanishing on a Cantor set][2]. 

  [2]: http://mathoverflow.net/questions/179445/non-zero-smooth-functions-vanishing-on-a-cantor-set