So, what does it mean for consciousness to be continuous? Is some brain activity enough, even if you're completely unconscious (i.e. anesthesia)? For that matter, if it wasn't continuous, would the current "you" be able to tell?

For that matter, if continuity of consciousness in general were an illusion, would the "you" at any given moment be able to tell? I guess the only reason it matters is that if there's no such thing as continuity, then nothing is lost upon upload (or death, really).