1. Current life is clearly possible (cogito ergo sum etc).
2. When this one ends, statistically it's inevitable there will be another one. See e.g. Boltzmann brain [1].
The pickle is that the timespan between 1 and 2 could be very long. But since you're not aware of it that doesn't matter. Also, the other pickle is you won't remember it, and that one really burns.
In other words, assert 1. and 2., therefore the human race does not die out when I die.
But, assert 1. and 2., therefore my life will continue seems like a pretty big leap to me.
There's an SF story where someone invents a teleporter, and it becomes very popular, but then one of them goes wrong and doesn't destroy the original body.
Sorry, I added the example of the SF story while you were replying to my post. In that story now there's two of the same person walking about, both of whom think they are 'him'.
The really disturbing this is, this might be what our lives are actually like. After all, our brains frequently stop being conscious and then summon consciousness back into being. Is it really the same consciousness though? Or is it a fresh newly synthesised process that just has memories from the last time your were conscious?
I still wouldn't use the teleporter though.
But if consciousness is based on what is percieved from physical world, then the consciousness can really take different forms, be parallel, be one single consciousness, and even not be dependant on time at all.
I think more interesting concept, than comparing death with being partially unconscious, is losing your complete memory. If consciousness is related on having past memories and experiences, how can we say a person can be conscious and function without having any single recollection of anything that ever happened.
And to push it further, in what sense is each moment's consciousness the same as the previous?
(I think there is a beaten up copy buried in my old closet at my mother's house, from when I was a boy.)