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To demonstrate more visibly the close relation between the continuity andintegrability, a new proof for the Banach-Zarecki theorem is presented on thebasis of the Radon-Nikodym theorem which emphasizes on measure-type propertiesof the Lebesgue integral. The Banach-Zarecki theorem says that a real-valuedfunction F is absolutely continuous on a finite closed interval if and only ifit is continuous and of bounded variation when it satisfies Lusin's condition(N).

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