  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 $\Phi$, dual to a class of meets" for which
"$\Phi$-continuous lattice" and "$\Phi$-algebraic lattice" are different
notions, thus for which a $2$-valued duality does not suffice.
