I have the following problem (taken from q1 p341 of Kolmogorov and Fomin's Introductory Real Analysis), which I am struggling to prove completely. I think I know how to show the only if part, but not the rest:
Prove that a function $f$ is absolutely continuous on $[a,b]$ if and only if it is a continuous function of bounded variation mapping every subset $Z \subset [a,b]$ of measure zero into a set of measure zero.
Any help would be greatly appreciated! Thank you :)