measure theory - Let $f$ be a continuous monotone function. Show that $f$ must be absolutely continuous on [0,1]

Let $f:[0,1]\to\mathbb{R}$ be a continuous monotone function such that $f$ is differentiable everywhere on (0,1) and $f'(x)$ is continuous on $(0,1)$. Show that $f$ must be absolutely continuous on $[0,1]$
and construct a counter example for the case without the "monotone" condition.

My attempt:

$f'(x)$ exists for all $x$ in $(0,1)$ anld continuous on (0,1), then $f'(x)$ is integrable on (0,1) and

$$f(x) = \int_{0}^{x}f'(x) + f(0),\forall x \in [0,1]$$

Since $f$ is an indefinite integral on [0,1] hence $f$ is absolutely continuous.

I want a different approach with the direct proof using the definition and not using so many theorems.

for the counterexample I have:

$f(x) = x\sin(1/x)$ and we must show that $f$ is not of bounded variation.