There is a problem with idea of continuity as a nature of identity, as some people suggested different meaning of it:

1. "Informational continuity" is when I remember in the moment N+1 my state of consciousness in the moment N. The problem: Every morning when I suddenly get up I have discontinuity in my memory stream as I loose memory of my dreams. Am I dead every morning?

2. Causal continuity. The main question here is continuity of what - of element of stream of conscuisness which creates next element in the next moment even if they are not remembered (so here arise difference with informational continuity idea), or causal continuity of underlying mechanism of brain. The main problem here will be that I will be not able (probably) feel if my causal stream was stopped and when restarted. (or i could feel it as one moment blackout which i will not remember?)