There's a movie out there called "The Ghost In The Shell", about a futuristic world where people regularly get cybernetic enhancements. Some people go so far as to replace their entire bodies, having their consciousnesses, whatever that is, moved into android bodies.
The logic behind this world is interesting. It states that there was a scientist who asked at what point a human stops being a human. If you lose an arm and have it replaced, are you not human? What about both arms? How about the torso? Eventually, he reasoned that there was never a point where a human would stop being human as long as it retained its consciousness.
But what is that? What is that thing that is transferred? Let's take this example out of the realm of fiction and move it into reality. How would such a process work? What, exactly, is "downloaded" into the android body? Most would say "thought patterns", but that begs the question - why would that remove the thought patterns from the original host? From everything we know, the information would be copied into the new form and deleted from the old. That means that, while it is functionally the same in every way, the android is not the same person. The original person died when the old thought patterns were deleted. That is, unless there's some kind of soul that you could move. I don't think that there is.
That thing that makes you who you are and differentiates you from an exact duplicate of yourself I will label Personal Identity. Soul carries too much baggage. When that personal identity ceases to exist, you are dead. That we will call "breaking the continuity".
This sort of thing happens all the time. If you're old enough to read and understand this, you don't have a single atom left from the body in which you were born. In many ways, you are a duplicate of your original. The continuity never seemed to break, though. There was never a moment where you ceased to be you and became a duplicate. Is it simply the gradualness of a process that determines if your personal identity remains or is lost?
I don't think so. That seems arbitrary. Illusory. I think that we do, slowly, over time, lose one personal identity and develop another. And another. But it's so gradual, so insignificant, that we'd never know. We are functionally the same. I have memories of myself as a little boy, but can I really say that I was that boy in the sense that I discussed? I don't know. Wouldn't an exact duplicate of me think the same thing, whatever the means of its construction?
Volumes of science fiction could and probably have been written about this subject. What about teleportation? Can we disintegrate a person and reconstruct her, atom by atom, in another place without breaking continuity? I doubt it. Wouldn't it be just as simple to analyze the atomic structure of that person and reconstruct her in that other place without disintegrating her? If you did, that other her would be functionally the same without sharing that same personal identity. Why, then, would we think differently if the first was annihilated? Because there's no one to claim otherwise, I suppose.
Going back to the first example, with cybernetics, let's say that you replace part of the brain with an electronic duplicate. Is it still the same person? Functionally, yes, but is the personal identity the same? I don't know. What if you gradually replace the entire brain with electronics, bit by bit over a period of years? Is the end product the same person you started with? While he or she may be exactly the same in every perceptible way, I would think not. I would think that this is a duplicate.
If this is the case, then why would we think differently of ourselves just because our brains are replaced atom by atom with organic material? There is no clear point at which continuity is broken, and yet it seems that it has been. What, exactly, is happening here?
All of these may very well be impossible to answer truthfully. This is one of the great laments of the rationalist. Because we refuse to use the easy, likely untrue answers, we often have to settle for no answer at all. If I do not allow for a soul that sits in the mass of atoms that composes my body and drives it the way I might drive a car, I can't know whether the child I see in old pictures of myself is actually me or if that's just how I remember it.