On Cantor sets every map is $C^{\infty}$ [closed]

Question

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

Answer 1

Let's see. $1/3 \in K$ is a limit point from the left of points in $K$. So try $$ g(x) = \left|x - \frac{1}{3}\right|^{1/2} $$ No function $f$ that agrees with $g$ on $K$ can have a finite derivative at $1/3$.