For a fixed Cantor set $K\subset [0,1]$ 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.

Suppose for example that $K$ is the middle third Cantor set.