And yet, it's interrupted every night when I sleep (at least, for some definitions of consciousness). I've been out for some surgeries, too, and I was still "me" when I came back. So I disagree with the "must be uninterrupted" idea.
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).
And, either after sleep or after anesthesia, I can tell that it wasn't continuous.
If continuity doesn't matter, then something still may be lost on upload - we won't know until we can interview someone who's been uploaded. And in death, if you don't believe in life after death, then something is definitely lost - the possibility of restart.