Hm.

My instinct is to say that I don't require causal continuity either... e.g., to say that if I appeared on Mars I would consider myself to still be the person who used to exist on Earth, despite the lack of causal connection.
That said, I don't really take that instinct seriously, since I'm incapable of actually imagining this happening without positing a causal connection I just don't happen to be aware of. The alternative is... well, unimaginable.

So, I dunno. Maybe I do require continuity in that sense, I'm just willing to posit it for any sufficiently complex system.