From this question, we know that a continuous function of bounded variation is not necessarily absolutely continuous. But the example (Devil's staircase) given is not differentiable. What if we require that the function is not only continuous but also continuously differentiable? Is every $C^1$ function that is BV absolutely continuous? Does it matter if we restrict the domain?
Sorry if these questions are too easy. I'm still getting used to these definitions.
Best Answer
This answer actually answers your question, too. (See also point (2) here.)
Yes. The antiderivative of an integrable function is absolutely continuous. If $f$ is $C^1$ and of bounded variation, then $\int \lvert f'\rvert = V(f) < \infty$. So $f$ is the antiderivative of an integrable function.