The answer "no" also follows by combining Propositions 79 and 92 in [Tao's lecture notes on differentiation theorems][1]. Indeed, let $f:[a,b]\to\mathbb{R}$ be an increasing differentiable function. By the quoted propositions, $f'$ is absolutely integrable, and
$$f(x)=f(a)+\int_a^x f'(t)\,dt,\qquad x\in[a,b].$$
Therefore, by a well-known criterion (cf. #6 of Exercise 87 in the notes), $f$ is absolutely continuous.


  [1]: https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/