Your doppelganger walks in the door just now. By asking probing questions, you establish he has a perfect copy of your memories and experiences, in every detail. He laughs at the same jokes, and cries at the same point of sappy movies you do (even though you never told anyone this). You can not discern any difference. Furthermore, he (apologies if you are female, just global replace the pronoun) has been engineered in a way that he will not age, or forget, or degrade in any way over time the way you will. Given the awkwardness of there being two people with one identity, he carefully explains that it only makes sense that he goes on as the "real you." To facilitate the process, he hands you a loaded gun, and tells you that you know what needs to be done and leaves the room.

Do you put the gun up to your head and pull the trigger? If not, why not? Surely, the fact of your memories going on is enough to assure you that you are not really "dying," just making it easier to resolve this complex situation?

Well, insofar as the doppelganger and I have sepata experiences, our personalities have already begun to diverge. Now, if the doppelganger whipped out a pair of EEG helmets and demonstrated that we could sync our experiences - eg, I take a sleeping pill and let my doppelganger go out on the town for a night, but I can easily 'catch up' the following morning - then i'd think about it.

I'd ask him to wait until I'm on my death bed and it's clear my death is unavoidable. Then I'd give him the thumbs up, instruct him to keep it real.

I'd argue that it matters whether the replacement is gradual or immediate.

There is really no difference for other people if the copy is perfect, but there is a huge difference for me.

Let's say I am able to make a perfect digital copy of myself, and immediately after I die. Well, my identity as a person has not in the least bit been affected because I have a perfect digital copy of myself. But the copy of my identity that was destroyed will never experience anything ever again, and that to me is the same as death.

However if the process is gradual, and the original copy of my identity experiences the process, then I would not view that as death.

Compare my identity to a computer program that you want to be moved to another location. You can either make a copy, and then delete the original, or you could copy one small piece at a time and update the file locations so that the program can be running while file locations are being changed. The only difference for you is that the second is probably a lot more work. But I'd argue that no complete copy was deleted in the second case. And if the program is sentient, I see this as being drastically different for that program.