  We develop a constructive theory of continuous domains from the perspective
of program extraction. Our goal that programs represent (provably correct)
computation without witnesses of correctness is achieved by formulating
correctness assertions classically. Technically, we start from a predomain base
and construct a completion. We then investigate continuity with respect to the
Scott topology, and present a construction of the function space. We then
discuss our main motivating example in detail, and instantiate our theory to
real numbers that we conceptualise as the total elements of the completion of
the predomain of rational intervals, and prove a representation theorem that
precisely delineates the class of representable continuous functions.
