Estevan Muguerza, Asier2020-05-292022-01-242020978-3-030-34225-82198-418210.1007/978-3-030-34226-5_5https://academica-e.unavarra.es/handle/2454/36989In 1956 R. D. Luce introduced the notion of a semiorder to deal with indifference relations in the representation of a preference. During several years the problem of finding a utility function was studied until a representability characterization was found. However, there was almost no results on the continuity of the representation. A similar result to Debreu’s Lemma, but for semiorders was never achieved. In the present paper we propose a characterization for the existence of a continuous representation (in the sense of Scott-Suppes) for bounded semiorders. As a matter of fact, the weaker but more manageable concept of ε-continuity is properly introduced for semiorders. As a consequence of this study, a version of the Debreu’s Open Gap Lemma is presented (but now for the case of semiorders) just as a conjecture, which would allow to remove the open-closed and closed-open gaps of a subset S ⊆ R, but now keeping the constant threshold, so that x + 1 < y if and only if g(x) + 1 < g(y) (x, y ∈ S).20 p.application/pdfeng© Springer Nature Switzerland AG 2020SemiordersDebreu’s Open Gap Lemmainfo:eu-repo/semantics/bookPartinfo:eu-repo/semantics/openAccessAcceso abierto / Sarbide irekia