We prove that the category of continuous lattices and meet- and directed join-preserving maps is dually equivalent, via the hom functor to $[0,1]$, to the category of complete Archimedean meet-semilattices equipped with a finite meet-preserving action of the monoid of continuous monotone maps of $[0,1]$ fixing $1$. We also prove an analogous duality for completely distributive lattices. Moreover, we prove that these are essentially the only well-behaved "sound classes of joins $Φ$, dual to a class of meets" for which "$Φ$-continuous lattice" and "$Φ$-algebraic lattice" are different notions, thus for which a $2$-valued duality does not suffice.

16 pages; revisions from refereeing