You need to more formally define consciousness and "continuity of consciousness" in order for this to be a worthwhile topic.

Whether it's a period of deep/no-dream sleep, a freeze and thaw cycle, emulation, or a copy operation, you're trying to answer the question of "what relationship do these two distinct-but-very-similar configurations of data (brain or electronic model) have to each other". Your brain at 2016-01-07T18:54:26,922604088 is different from your brain at 2016-01-07T18:54:55,035292477, but in many ways, you want to treat them as "the same".

Define the critera for identity, and this question will dissolve.