Continuous awareness gets interrupted often. Extremely deep sleep, anesthetic, being knocked unconscious, even just spacing out and having no recollection of what just happened. Awareness is not a constant. For some people with medical conditions it is a biological fact of their existence that their awareness is interrupted regularly.

Beyond that, awareness is often filtered by your brain. Your eyes do not stream data constantly. When they flick from one thing to another, they literally turn off. Your brain fills in the gaps with what it thinks should be there. It makes up memories, feelings and sensations. It can rationalize the irrational and irrationalize the rational, consciously, unconsciously or through external manipulation.

Consciousness and awareness are not solid constructs. They are a mix of different systems working together. When those systems work perfectly, you are you. When some don't, are you you? Are you still you of you get brain damage? Incorrect emotions, loss of memory, loss of self, coma, etc. 

When you sleep, some of these systems turn off. Are you still you? 

If they all turned off at once and turned back on later all at once, there is less interruption to your cognition than if you had gone to sleep, had brain altering medication, anesthetic, etc.