Let $f\colon[a,b]\to\mathbb R$ be an increasing differentiable function. Then 
$$f(x)-f(a)=(HK)\int_a^x f'(t)\,dt$$
for all $x$,
where $(HK)\int$ is the [Henstock–Kurzweil integral][1]. Therefore and because $f'\ge0$, $f'$ is Lebesgue integrable and hence 
$$f(x)-f(a)=(L)\int_a^x f'(t)\,dt$$
for all $x$,
where $(L)\int$ is the Lebesgue integral. 

So, $f$ is absolutely continuous. 

  [1]: https://en.wikipedia.org/wiki/Henstock%E2%80%93Kurzweil_integral#Properties