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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 ><

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1 Answer 1

up vote 3 down vote accepted

EDIT: In response to the objection raised in the comments, I think this works:

Since $K$ is compact, its complement in $[0,1]$ is open, thus a countable union of open intervals. Enumerate them any way you like. For $x$ in the closure of interval $I_k$, define $f(x)$ to be $\sum2^{-j}$ where $j$ runs over all those numbers such that $I_j$ is to the left of $I_k$.

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How will you enumerate the rationals so that you obtain continuity and monotonicity? Just say it so? I like your idea though, let me think about it for a little bit. –  Taylor Martin Jun 17 '12 at 1:33
    
Monotonicity is not a problem. The open intervals don't overlap, so if interval $I$ is left of interval $J$, then $f(x)\lt f(y)$ for all $x$ in $I$ and all $y$ in $J$, and that carries over to the points in $K$. But continuity, that's a problem. I'll leave my answer up so maybe someone can use it as a starting place, but I think continuity is an issue. –  Gerry Myerson Jun 17 '12 at 1:56
    
I might have a better idea. See post. –  Taylor Martin Jun 17 '12 at 3:00
    
Thank you Gerry ~ This is much simpler than my idea! –  Taylor Martin Jun 17 '12 at 4:01

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