Consider sleep. The consciousness that goes to sleep ends. There is a discontinuity in perceived time. In the morning, the wakening brain ...

[...] will be capable of generating a perfectly functional consciousness, and it will feel as if it is the same consciousness which observes the mind which is, for instance, reading these words; but it will not be. The consciousness which is experiencing awareness of the mind which is reading these words will no longer exist.

You cease to exist every night. Indeed, there are all sorts of disruptions to the continuity and integrity of consciousness, ranging from distraction to coma to seizures to dissociative drugs. But people who experience these still care about their future selves. Why? Are they in error to do so?

My point here is that we can argue "this consciousness ceases to exist" with about as much strength for sleep as for more exotic processes. The difference is social and psychological, not metaphysical: we are accustomed to sleep, and to treating the consciousness who is born in the morning as the same consciousness who died the night before. It makes sense socially to do so; it is adaptive to do so; it is certainly more conducive to an intuitive understanding of things like memory.

But sameness — identity — is pretty darn tricky. Electrons don't have it; where does it come from?