Does $f$ absolutely continuous imply $|f|$ absolutely continuous?

Question

$f: [a,b] \to \mathbb{R}$ is absolutely continuous $\Leftrightarrow$ $$\forall \epsilon>0 \ \exists \delta>0: \sum_{i=1}^{n}(b_i-a_i)<\delta \Rightarrow \sum_{i=1}^{n}|f(b_i)-f(a_i)|<\epsilon$$

where $(a_i,b_i)$ are disjoint subsets of $[a,b].$

Does $f$ absolutely continuous imply $|f|$ absolutely continuous? And what about $\frac{1}{|f|}$?

Answer 1

Hint

Use the inequality $$||x|-|y||\le |x-y|.$$