  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 finite closed interval if and only if
it is continuous and of bounded variation when it satisfies Lusin's condition
(N).
