This paper introduces the frame of partially defined real numbers and the lattice-ordered ring of partial real functions on a frame. This is then used to construct the order completion of rings of pointfree continuous real functions. The bounded and integer-valued cases are also analysed. The application of this pointfree approach to the classical case C(

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Aitziber Etxezarreta-Etxarri a (https://orcid.org/0000-0002-4239-092X) Julen Izagirre-Olaizola b* (

This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case) and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth's classical construction to the pointfree setting. The second shows that in the bounded case the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L and that in the non-bounded case it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov's approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.