Problem 4

Exercise 11.4 (Absolute continuity)  

1. Let $f$ and $g$ be absolutely continuous functions on $[0,1]$. Show that their product is also absolutely continuous.
2. Give an example of a function on $[0,1]$ which is uniformly continuous but not absolutely continuous.

Proof. Part (a) is proven in Exercise .

For part (b), consider the Devil's staircase. It is a continuous function on a compact set and thus uniformly continuous. But an absolutely continuous function has bounded variation, which the Devil's staircase does not, so we know the function is not absolutely continuous by contradiction. $\qedsymbol$