Before you were born you didn’t have the thing you call a consciousness. But now you do.
I think continuity is a red herring, realistically our whole idea of consciousness is probably off base. It might not even be a real thing, for example my pet cat doesn’t care if he has a consciousness or not because he never invented a word for it.
Perhaps I do. Perhaps it's only there when I pay attention to it, and my brain smooths over the gaps like I know it does with saccades and blind spots (both literal and metaphorical). And even if it is there all the time, is it like yours?
I think continuity is a red herring, realistically our whole idea of consciousness is probably off base. It might not even be a real thing, for example my pet cat doesn’t care if he has a consciousness or not because he never invented a word for it.