Defining the Cantor set $C$ in the typical way, and $O=[0,1]\setminus C$, where would the Cantor-Lebesgue function $\varphi$ map $O$?
I believe that the image of $\varphi(C)$ is the entire interval $[0,1]$ (is this correct?), so what would that mean for the image of $\varphi(O)$?
Additionally, $\varphi$ has been defined in the traditional way on $O$, and has been extended to all of $[0,1]$ via $\phi(x)=\sup\{\varphi(t)|t\in O\cap [0,x)\}$.