Some longer works were published a part at a time, over several issues. A novel might run to eight or ten parts. This site keeps them together.
The bar at the top of a part tells you which work you are reading and how far through it you are, for example "Part 3 of 8". Tapping Contents in that bar shows you the whole list.
At the foot of a part there are two cards, one for the part before and one for the part after, each showing its title so you know what is coming. On a phone there is also a small bar along the bottom of the screen with Previous, Contents and Next, so you never have to scroll back up to move on.
The contents page lists every part in reading order, numbered. That order is the order they were written in, not the order of their dates: some parts have no recorded date at all, and where a date does exist it is the date of the magazine rather than the place of the part in the story.
What "Continue" means
Once you have opened a part, the contents page marks it Continue, and the button at the top changes from "Start reading" to "Continue reading" and takes you back to it. It is simply a note of the last part you opened, so you can find your place again without remembering the number.
It is kept on your own device and nowhere else. Nothing about your reading is sent to us or to anybody else. We do not know what you have read, and we have not built any way to find out.
That is true whether or not you have signed in. Some of the newest writing asks for an email address before it can be read in full, and signing in does nothing but open those pages: it does not follow you around the site and it does not record where you have got to in anything. Your place in a column is a note your own browser keeps.
Two things follow from that, and neither is a fault. If you read on your phone and then on a computer, each one remembers its own place, because they are two different devices. And if you clear your browsing data, or your browser is set not to keep anything, the mark disappears and every part is simply listed as it was. Nothing else changes, and nothing is lost: the parts are all still there, in order.