(Not a frequent poster here, but nobody's responded yet and Foucault was the author who I read most in college, so mods please forgive me or let me know how I can improve this if it doesn't meet standards).

Oh boy, I actually kind of enjoyed Archaeology of Knowledge, so hopefully I can help. Foucault was routinely criticized for not having a specific and apparent methodology to explain how he was able to reach his conclusions in some of his earlier works on prisons and hospitals, for example, so he wrote Archaeology as a sort of response and also laid out some of his theories on historiography. He borrowed heavily from Nietzsche's concept of genealogy in The Genealogy of Morals (if you aren't familiar, the basic argument was that the terms "good" and "bad"/"evil" are loaded with connotations based on how they were used historically--he traces the origins of the term and explains that there was a relationship between "goodness" and the ruling class because it was presumed their status was a result of their being "good", and that this connotation never actually left our conception of goodness even if the use of the term changed over time.)

Foucault built upon this concept further and developed it into his concept of archaeology as a historical methodology. I actually strongly prefer the way it's explained in Foucault by Gilles Deleuze--Deleuze uses map-making analogies to describe how Foucault viewed history and discourse (I promise this is leading up to an answer to your question). He describes discourse over time as a sort of topology; the "map" itself consists of all possible things which could be said in a discourse on a certain topic. Over time, as regimes change, knowledge and discourse begin to "accumulate" in certain areas (what we might imagine to be a higher elevation on a topographical map) or new areas may be added to the map.

As it concerns Foucault's theory, the concepts of continuity and discontinuity are meant to reflect the "flow" of history and how "things are no longer perceived, described, expressed, characterised, classified, and known in the same way," as he describes it in Archaeology. He was looking at how a culture or society generates an episteme which is basically the configuration of knowledge which then determines social orders and practices. Over time, certain "things" will have continuity as societies figure out what constitutes truth and knowledge while other "things" will experience discontinuity during the transition from one era or regime into the next.

I think a good example to illustrate this would be Thomas Kuhn's The Structure of Scientific Revolutions (though I want to caution that I'm not implying these two would necessarily agree with each other). Kuhn argues against the notion that there is a sort of natural "progress" which occurs in the realm of science as knowledge and facts accumulate over time, and argues instead that science is actually conducted under certain paradigms which are assumed to be true until some anomaly is encountered which throws the paradigm into question and begins a "revolution" which eventually leads to a paradigm shift. He gives as an example the shift from the Ptolemaic system which had Earth as the center of the universe to the heliocentric model of Copernicus. For Foucault, this paradigm shift from an episteme which assumes an Earth-centered universe to a heliocentric one represents a discontinuity.