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As discovered recently, Li and Wang's 1997 treatment of semicontinuity for frames does not faithfully reflect the classical concept. In this paper we continue our study of semicontinuity in the pointfree setting. We define the pointfree concepts of lower and upper regularizations of frame semicontinuous real functions. We present characterizations of extremally disconnected frames in terms of these regularizations that allow us to reprove, in particular, the insertion and extension type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H. Li [Algebra Universalis 44 (2000), [271][272][273][274][275][276][277][278][279][280][281] in the right semicontinuity context. It turns out that the proof of the insertion theorem becomes very easy after having established a number of basic results regarding the regularizations. Notably, our extension theorem is a much strengthened version of Li and Li's result and it is proved without making use of the insertion theorem.

We study necessary and sufficient conditions for the continuous Scott-Suppes representability of a semiorder through a continuous real-valued map and a strictly positive threshold. In the general case of a semiorder defined on topological space, we find several necessary conditions for the continuous representability. These necessary conditions are not sufficient, in general. As a matter of fact, the analogous of the classical Debreu's lemma for the continuous representability of total preorders is no longer valid for semiorders. However, and in a positive direction, we show that if the set is finite those conditions are indeed sufficient. In particular, we characterize the continuous Scott-Suppes representability of semiorders defined on a finite set endowed with a topology.

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In this paper, we go further on the problem of the continuous numerical representability of interval orders defined on topological spaces. A new condition of compatibility between the given topology and the indifference associated to the main trace of an interval order is introduced. Provided that this condition is fulfilled, a semiorder has a continuous interval order representation through a pair of continuous real-valued functions. Other necessary and sufficient conditions for the continuous representability of interval orders are also discussed, and, in particular, a characterization is achieved for the particular case of interval orders defined on a topological space of finite support.