Ytukyg

I have been using Real Analysis by Royden and Fitzpatrick. I am currently stuck on problem 39 of chapter 6. The problem is stated as follows:

"Use the preceding problem to show that if f is increasing on [a, b], then f is absolutely continuous on [a, b] if and only if for each $epsilon$ , there is a $delta$ > 0 such that for a measurable subset
E of [a, b],
m*(f(E))< $epsilon$ if m(E) <$delta$"

(m* denotes outer measure and m denotes lebesgue measure)

I can see why absolute continuity implies the other property. However, I fail to see how absolute continuity follows from the fact that f is increasing and that for each $epsilon$ , there is a $delta$ > 0 such that for a measurable subset
E of [a, b],
m*(f(E))< $epsilon$ if m(E) <$delta$"

My approach is to take $delta$ responding to $epsilon$. Then for any finite collection of disjoint open interval ($a_k$,$b_k$) such that $sum_{k=1}^n (b_k-a_k) < delta$ we have that m*$(f(cup_{k=1}^n(a_k, b_k)))<epsilon$. I cannot see how this implies $sum_{k=1}^n |f(b_k)-f(a_k)|<epsilon$ as required for absolute continuity.

Is this a good approach to solve this problem? Any advice would be appreciated!

EDIT:

The previous problem (Q38):

Show that f is absolutely continuous on [a, b] if and only if for each $epsilon$ > 0, there is a $delta$ > 0 such that for every countable disjoint collection ${(a_k, b_k)}$ of open intervals in (a, b), if $sum_{k=1}^infty |b_k-a_k|<delta$ then $sum_{k=1}^infty |f(b_k)-f(a_k)|<epsilon$