I'm trying to prove that for any $K\subset[0,1]$ such that K is compact and $m(K)=0$ with no isolated points, there exists a continuous, monotonic increasing function $f$ which maps $[0,1]$ onto $[0,1]$, and such that $f'(x)=0$ for every $x\in[0,1]-K$ (i.e. $f'=0$ a.e. $x$).
Now, it is clear that the Cantor-Lebesgue function $F$ will do for $K=\mathscr{C}$, and a modification of $F$ will accommodate any Cantor-like set (sets of constant dissection).
I'm wondering if there are any other sets $K$ with the properties listed. If so, can they be narrowed down into cases and a constructive proof applied to each case?
If not, then how does one go about proving the statement for general $K$?
PROGRESS:
Here's my current idea.
Let $K$ be as above. Define $K_{0}=[\inf K, \sup K]$. Dissect the middle third of $K_{0}$ into pieces $J_{1}^{1}$ and $J_{1}^{2}$ by removing the open middle segment. Define $K_{1}:=[\inf K, \sup K\cap J_{1}^{1}]\cup[\inf K\cap J_{1}^{2}, \sup K]$. Then dissect this set by removing the middle thirds of the two component intervals, yielding $J_{2}^{1}$, $J_{2}^{2}$, $J_{2}^{3}$, $J_{2}^{4}$. Then put $K_{2}=[\inf K, \sup K\cap J_{2}^{1}]\cup[\inf K\cap J_{2}^{2}, \sup K\cap J_{2}^{2}]\cup[\inf K\cap J_{2}^{3}, \sup K\cap J_{2}^{3}]\cup[\inf K\cap J_{2}^{4}, \sup K]$. Then continue dissecting $K$ as such. The $\sup$ and $\inf$ appearing in each definition exist since every $J$ is compact, and the intersection of two compact sets is again compact.
Now, at the jth step of the construction, we have $K_{j}$ being the union of $2^{j}$ component subintervals. On these subintervals, simply require $f$ to be linearly increasing, and on the corresponding open subinterval which is removed, define $f=\frac{n}{2^{j}}$ where $n$ denotes the nth subinterval going left to right. Then $f$ evaluated at the least element of $K$ is $1$ (if we begin by $n=0$) and $f$ evaluated at the largest element of $K$ is $1$, and the function is monotonic increasing.
I guess now I'm running into the same problem...establishing continuity. I want to say something like \begin{equation*} |f_{j+1}(x)-f_{j}(x)|<\frac{2}{j} \end{equation*} so that I can establish uniform convergence.
These Cantor problems never get any easier ><