Our subjective experience of the 'continuity' of consciousness is surely an illusion. But what makes that illusion and why is it so compelling? That's a deep question but here are I think two fundamental reasons:
1. Embodiment. You are an embodied intelligence. It is a mistake to think of mind and body as somehow separate. Our conscious experience and its awakening as a developing child is surely deeply rooted in our physical experience of the world as mediated through our senses.
2. Environmental continuity. Our experience of the world changes 'relatively' slowly. The word relative is important here since I mean relative to the rate at which our conscious experience updates itself. Of course we do experience the discontinuity of going to sleep then waking to the changed world of a new sunrise, but this is both deeply familiar and predictable.
A word that encompasses both of these is situatedness. Our intelligence, and hence also conscious experience is inextricably situated in our bodies and in the world. Let me illustrate what this might mean with a thought experiment.
Imagine a brain transplant. Your brain complete with its memories and life's experience, together with as much of your central nervous system as might be needed for it to function properly were to be transplanted into a different body. You would wake from the procedure into this new body. I strongly suspect that you would experience a profound and traumatic discontinuity of consciousness and, well, go mad. Indeed it's entirely possible that you simply couldn't (and perhaps mercifully) regain or experience any sort of consciousness at all. Why? Because the conditions for the emergence of consciousness and the illusion of its continuity have been irreversibly broken.
However, if what I have said above is true, there's a flip side to the story that could have extraordinary consequences.
If the continuity of consciousness is an illusion then, in principle at least, it might be possible to artificially perpetuate that illusion.
Imagine that at some future time we have a sufficiently deep understanding of the human brain that we can scan its internal structures for memories, acquired skills, and all of those (at present dimly understood) attributes that make you you. It's surely safe to assume that if we're able to decode a brain in this way, then we would also be able to scan the body structures (dynamics, musculature and deep wiring of the nervous system). It would then be a simple matter to scan, at or just before the point of death, and transfer those structures into a virtual avatar within a virtual world. The simulated brain structures would be 'wired' to the avatar's virtual body in exactly the same way as the real brain was wired to its real body, thus satisfying the requirement for embodiment. If the virtual world is also a high fidelity replica of the real world then we would also satisfy the second requirement, environmental continuity.
I would argue that, under these circumstances, the illusion of the continuity of consciousness could be maintained. Thus, on dying you would awake (in e-heaven), almost as if nothing had happened. Except, of course, that you could be greeted by the avatars of your dead relatives. Even better, because e-heaven is just a virtual environment in the real-world, then you could just as easily be visited by your living friends and relatives. Could this be the retirement home of the far future?
In this way human consciousness could, I believe, be immortal.