  In this paper, infrared regularization of semi-infinite entangling regions
and island formation for regions of finite size in the eternal Schwarzschild
black hole are considered. We analyze whether the complementarity property and
pure state condition of entanglement entropy can be preserved in the given
approximation. We propose a special regularization that satisfies these two
properties. With regard to entangling regions of finite size, we derive two
fundamental types of them, which we call "mirror-symmetric" (MS) and
"asymmetric" (AS). For MS regions, we discover a discontinuous evolution of the
entanglement entropy of Hawking radiation due to finite lifetime of the island.
The entanglement entropy of matter for semi-infinite regions in two-sided
Schwarzschild black hole does not follow the Page curve. The lifetime of AS
regions is bounded from above due to the phenomenon that we call "Cauchy
surface breaking". Shortly before this breaking, the island configuration
becomes non-symmetric. For both types of finite regions, there is a critical
size, below which the island never dominates. For regions smaller than some
other critical size, the island does not emerge. Finally, we show that the
island prescription does not help to solve the information paradox for certain
finite regions.
