Recall that the famous Cantor-Lebesgue function $\phi (x)$ is a continuous, increasing surjection from $[0,1]$ to $[0,1]$ with the property that $\phi (0)=0$, $\phi(1)=1$ and derivative equal to $0$ everywhere outside the Cantor set.
It is well known that the Cantor set is uncountable and has Lebesgue measure zero. The question is whether or not such a function $f:[0,1]\to \mathbb{R}$ with the same properties as the Cantor-Lebesgue function can be constructed (or exists) for an arbitrary uncountable set $E$ with Lebesgue measure zero, meaning that we replace the Cantor set with ANY uncountable set of measure zero.
A possible approach is to show that any uncountable set has a closed, perfect subset in which case the same construction as Cantor-Lebesgue function can be done. Any suggestions?
