I have been working on a problem for a while and ultimately I would like to prove that a certain function is absolutely continuous, strictly monotone increasing, and its derivative vanishes on a set of positive Lebesgue measure. Now, I thought I had this question solved but I was being naive. The function in question is constructed as follows, $F \subseteq[0,1]$ is a fat Cantor set with positive measure. Let $E = [0,1] \setminus F$, then for every $x \in [0,1]$ define $$f(x)= \int_0^x\mathbb{1}_E dt.$$ Now I think proving that this is absolutely continuous and that its derivative vanishes on a set of positive measure is independent of $E$ and $F$, please see here for an argument I gave earlier today to justify the latter. However proving that this is strictly monotone increasing is what is bothering me now. I had a mistake in my logic when working on this problem, but I cannot seem to form the correct argument. Perhaps here we do make use of the fat Cantor set.
Does anyone have any idea how to prove that $f$ is strictly monotone increasing?
Thank you so much, sorry for the many questions!
Krull.