Debreu's open gap lemma for semiorders

Abstract

The problem of finding a (continuous) utility function for a semiorder has been studied since in 1956 R.D. Luce introduced in \emph{Econometrica} the notion. There was almost no results on the continuity of the representation. A similar result to Debreu's Lemma, but for semiorders, was never achieved. Recently, some necessary conditions for the existence of a continuous representation as well as some conjectures were presented by A. Estevan. In the present paper we prove these conjectures, achieving the desired version of Debreu's Open Gap Lemma for bounded semiorders. This result allows to remove the open-closed and closed-open gaps of a subset $S\subseteq \mathbb{R}$, but now keeping the constant threshold, so that $x+1<y$ if and only if $g(x)+1<g(y) \, (x,y\in S)$. Therefore, the continuous representation (in the sense of Scott-Suppes) of bounded semiorders is characterized. These results are achieved thanks to the key notion of $\epsilon$-continuity, which generalizes the idea of continuity for semiorders.