If $f$ is continuous over $[a,b]$ and $\vert f\vert$ has bounded variation, is $f$ absolutely continuous?
Given $\varepsilon >0$. I need to find a $\delta$ such that $\sum\vert f(b_i)-f(a_i)\vert<\varepsilon$ when $\sum(b_i-a_i)<\delta,$ but this doesn't use the fact that $\vert f\vert $ has bounded variation.