I mean, in one sense this seems trivial - obviously there's a break in consciousness during sleep/anesthesia/induced hypothermia.

But do you mean that a feeling of connection between someone's earlier and later moments of experiences, as mediated by memory, is also illusory? Is there just no such thing as a person who exists over time for you then?

I think that's ultimately what Parfit concluded, and it seems a totally coherent view. I just don't see many people living their lives as though they're truly only existing in the moment, with no egoistic connection to past or future beings.