By continuity of identity I mean the classic teleportation problem: when you teleport, do you arrive at the destination, or do you die and your clone arrives? The problem being that no-one other than you can tell; and by the time you know, you're dead.
In EP it seems reasonable to assume that no one would ever want to resleeve if it caused their death or if it might do so. That implies that in the setting there is some way in which it was shown, when the technology was invented or before it passed into widespread use, that resleeving is safe. Yet it is not clear what this was or what other effects it had on the setting.
I know this question has been asked before in terms of the soul but this is not the same metaphysical question because it is practical: life or death. Regardless of the philosophical nature of the question noone would want to go through a process that is not proven to have a chance of survival. In fact proving a soul exists might have been the proof of safety.
How do Eclipse Phase citizens know that resleeving does not kill them and create a clone?
Edit: just to clarify, I'm not asking how resleeving works in the setting. Rather, I'm asking this: suppose that one day Professor Q was to demonstrate the first ever resleeving machine. He steps in and then comes out in a new morph. Then a heckler shouts, "That's not Professor Q, it's a clone of him with his memories! The original is dead! He might think it worked, but that's because the clone has his memory of getting into the machine. It might look exactly the same to him and everyone else, but the man who stepped into the machine is now dead forever, and if you use that machine, you'll die too!"
What can Professor Q show the audience to prove the heckler wrong? If the answer is "nothing", why would anyone accept and use the machine? If we assume that he did show them something that proved it, then what was it, and what other implications does its existence have for the setting? As far as I know, none of the books address this, and whichever way it resolves has pretty major implications for the tone of the world.
Best Answer
They don't
One of the biggest hurdles to overcome when resleeving is the shock caused by the loss of continuity (p272):
Fortunately, this doesn't always happen. When resleeving under normal conditions, the mind is gradually transferred via an ego bridge (p269). Parts of it "run" in both brains at once as the process goes on, and the person can even be awake at the time. In this case, it can be said that there is no "true" self left behind.
While this particular concern is not mentioned as part of the bioconservative agenda, it seems like that's where it would belong, as they oppose uploading. The beauty of Eclipse Phase is that it encourages you to examine such questions.