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Having been back and forth on this issue a lot over the last few years, I've lost any confidence in the validity of the question as to where the self begins and ends. I can find no logically sound test that indicates whether individual identity under a given case is continuous or not. It seems to ultimately come down to a cultural semantic argument, and the closest cognitive science can come is that the "self" is a projection, convenient within a brain and across them for labelling, tracking and predicting behaviour patterns.
Growing from an infant into a child into an adult; having the body replace its own cells and/or chemicals; becoming unconscious; brain damage; near death experiences; religious rebirth; spiritual experiences; disassociative states; being ridden; psychopharmaceuticals; losing a limb; abiotic prosthetic implants; organ transplants; tissue scaffolded pseudotransplants; cranial magnetic interference; bundle theory; it's the ship of Theseus by a thousand names. Each argument describing a true core self as something other than opinion or tradition strays either into the ad hoc or unfalsifiable metaphysics.
That's not to say that there is not a "self," but there is no way of deliniating where it begins and ends that holds up to empirical testing. It makes the question "does God exist" seem simple by comparison.