TITLE: Is there an increasing function on $[a, b]$ which is differentiable, but not absolutely continuous？
QUESTION [7 upvotes]: Is there an increasing function on
$[a, b]$ which is differentiable,
but not absolutely continuous？

REPLY [5 votes]: Such a function does not exist.
Indeed, 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. In particular, $f$ is Henstock–Kurzweil integrable on $[a,b]$.
Therefore and because $f'\ge0$, $f'$ is Lebesgue integrable on $[a,b]$, and hence
$$f(x)-f(a)=(L)\int_a^x f'(t)\,dt$$
for all $x$,
where $(L)\int$ is the Lebesgue integral.
So, $f$ must be absolutely continuous.