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In this paper, infrared regularization of semi-infinite entangling regionsand island formation for regions of finite size in the eternal Schwarzschildblack hole are considered. We analyze whether the complementarity property andpure state condition of entanglement entropy can be preserved in the givenapproximation. We propose a special regularization that satisfies these twoproperties. With regard to entangling regions of finite size, we derive twofundamental types of them, which we call "mirror-symmetric" (MS) and"asymmetric" (AS). For MS regions, we discover a discontinuous evolution of theentanglement entropy of Hawking radiation due to finite lifetime of the island.The entanglement entropy of matter for semi-infinite regions in two-sidedSchwarzschild black hole does not follow the Page curve. The lifetime of ASregions is bounded from above due to the phenomenon that we call "Cauchysurface breaking". Shortly before this breaking, the island configurationbecomes non-symmetric. For both types of finite regions, there is a criticalsize, below which the island never dominates. For regions smaller than someother critical size, the island does not emerge. Finally, we show that theisland prescription does not help to solve the information paradox for certainfinite regions.

Infrared Regularization and Finite Size Dynamics of Entanglement Entropy in Schwarzschild Black Hole