# Continuously differentiable functions of bounded variation

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.

## 2 Answers

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.

• You are welcome. Commented Mar 16, 2013 at 14:27

You don't even need to require bounded variation. Every $C^1$-function is absolutely continuous by the fundamental theorem of calculus: The derivative exists everywhere and integrates back to the original function (even by Riemann integration!).

• The function $x \mapsto x^2$ is not absolutely continuos on an unbounded interval. Commented Mar 14, 2013 at 16:50
• Fair enough, you are right. I was thinking of a compact interval. But I think if you require bounded variation, then it works even for unbounded domains. Commented Mar 15, 2013 at 10:12