[sunaurus]

But why would the feeling of continuity be broken by an instant replacement?

In both cases (gradual and instant replacement), at any given moment, there exists only one entity with your consciousness and self of sense and continuous memories leading up to that moment.

[martindbp]

I think my point is, maybe there's no such thing as continuity of experience (since we sleep every day) and consciousness itself may be an illusion. With an instant replacement, and also with a large distances, it makes it clear this is not me any longer, but if I replace my neurons gradually with digital components it's essentially equivalent to what is already happening with my brain cells, so it doesn't break the illusion. My identity and consciousness is an illusion and I just don't want to break it. I know that is contradictory and makes no sense, but that's the reason I would not use the transporter.

[lores]

It's not a replacement, a collection of atoms is assembled on a remote planet. And it needs not be a single entity, as the thought experiment with the failure to destroy the original body shows.