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

Constructive Domains with Classical Witnesses