Continuous function is absolutely continuous on [0,1] if it is increasing

Question

Let $f$ be a continuous function on $[0,1]$ that is absolutely continuous on $[\epsilon, 1]$ for $\epsilon \in (0,1)$.

a. Show that $f$ may not be absolutely continuous on [0,1]
b. Show that $f$ is absolutely continuous (AC) on $[0,1]$ if it is increasing

This question is chapter 6.37 from Royden-Fitzpatrick Analysis. I used a counterexample for (a).

As for (b), my reasoning is as follows.

1. Let $f$ be a monotone increasing function on $[0,1]$.
2. A monotone function $f$ is the sum of an absolutely continuous function $g$ and a singular function $h$, so $f = g +h$.
3. $f' = g' + h'$ and since $h$ is singular, $h'=0$ almost everywhere, so $f'=g'$ on $[0,1]$.
4. Since $f$ is AC on $[\epsilon,1]$, $f = g $ on $[\epsilon,1]$ implies $h \equiv 0$.
5. Therefore $f = g$ on $[0,1]$, so $f$ is AC.

Is the argument in step 4 legit? Thanks for any help.