Nearly all web APIs get paging wrong

I'd like to see this continuation pattern described in a bit more detail. How does the client define the sort parameters or preferred data limits per continuation? Have I missed the point?

I've often wondered how the paging on HN could be better. The main issue is going from page 1 to page 2 where items move between them and I either see items a second time, or miss them. The problem with fixing a sequence on first page load is then when to refresh for new content--only on page 1? Lastly, a prescription is not helpful without a design for efficient implementation. How can this be achieved in a stateless manner?