Analysis: Absolute Continuity & Differentiation

Let A $\displaystyle \subset $[0, 1] be a Borel set such that 0 < m(A $\displaystyle \cap $ I) < m(I) for all interval I $\displaystyle \subset $ [0, 1]. Let

F(x) = m([0, x] $\displaystyle \cap $ A), where m is lebesgue measure. Show that:

1. F(x) is absolutely continuous and strictly increasing on [0, 1]

2. F'(x) = 0 on a set of positive measure.

I've shown 1, but can't prove 2. In fact, 2 would seem to be false because if F is Abs Cont and F' is zero on a set E, then $\displaystyle F = \int_E F' $ implies that F is 0 on the set E. But, wouldn't this contradict the fact that F is strictly increasing?