I assume the most basic scenario that we're just complex patterns of self referencing data (electrical activity), your conciousness can cease during anethstetic or some types of coma or rare cases of temporary brain death and start right up again no problem, if someone replicates a close enough copy of you then that's got an equal claim to your continuity as the you that wakes up in the morning, even if you're still alive both of the yous are you, the yous could even be merged again so long as no data is destroyed the new hybrid you would have a completely valid claim to both continuities that emerged following the intitial seperation.
TLDR: You're no more special than a videogame save file and that's fantastic because it makes it really easy to be immortal or get resurected.