Continuity of consciousness very quickly becomes a hard word to define : not only do you interrupt consciousness for several hours on a nightly basis, you actually can go into reduced awareness modes on a regular basis even when 'awake'.

Moreover, it might not be necessary to interrupt continuity of consciousness in order to "replace parts" in the brain. Hemispherectomies demonstrate that large portions of the brain can be removed at once without causing death, for example.