A new proof for the Banach-Zarecki theorem: A light on integrability and continuity

To demonstrate more visibly the close relation between the continuity and integrability, a new proof for the Banach-Zarecki theorem is presented on the basis of the Radon-Nikodym theorem which emphasizes on measure-type properties of the Lebesgue integral. The Banach-Zarecki theorem says that a real-valued function F is absolutely continuous on a nite closed interval if and only if it is continuous and of bounded variation when it satises Lusinʹs condition. In the present proof indeed a more general result is obtained for the Jordan decomposition of F.