What if you replaced your brain piecemeal, each replacement component interacting with your analog brain until there's none left. Still not you?
That said, it could be argued that this sort of thing happens all the time anyway. I'm not the person I once was. In a sense, the old me died and was replaced by the new me. Yet I still consider the new me to be myself.
Or maybe not. The child me is dead, though echoes of him may live on in me.
I guess what disturbs me about a mechanical/cybernetic brain replacement is that it would feel to me too much like some technological virus or disease taking over my mind and body -- even if it gave me some super powers or preserved me in some way. I guess some injured people or people who are dying anyway might not have much of a choice.. but as long as I still have a choice I think I'd choose to remain organic.
It's not an easy choice, however, and I would have no problem use various prosthesis like glasses.. or enhancing my memory with external devices like smartphones/PDA's/computers. It's just too invasive for that to be made a part of me... at least if it starts to make decisions. But probably children born in to a world where that sort of thing was commonplace probably wouldn't have much of a problem with it.
They don't necessarily. Some people do feel like they're not themselves after organ transplants. Like their hand isn't theirs after a hand transplant.. or that it has a mind of its own. Some experience personality changes or feel like the new body parts have their own desires.
Oliver Sacks and other neurologists and psychologists have written about and studied this phenomena.
Gradual brain replacement might have none of these issues, or it might have them, or even worse ones, or perhaps the changes will be more insidious and not noticeable by the people who they have them but could be noticeable by others.
Even in the best case scenario, when they're without these neurological or psychological issues, that would not clear them on the philosophical level. The questions of identity would remain.
The uploaded consciousness or gradual brain replacement are pretty different from this case, though, it's true. So maybe these issues wouldn't exist or not exist in the same way, but philosophical issues of identity would remain.