a new proof for the banach-zarecki theorem: a light on integrability and continuity

to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuous and of bounded variation when itsatisfies lusin's condition. in the present proof indeed a moregeneral result is obtained for the jordan decomposition of $f$.