Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof |
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| Institution: | 1. Dipartimento di Matematica Applicata “Bruno de Finetti”, Universitá di Trieste, Piazzale Europa 1, 34127 Trieste, Italy;2. Department of Economics, University of Queensland, Brisbane, Qld 4072, Australia;1. Department of Mathematics, the Pennsylvania State University, University Park, PA 16802, United States;2. Department of Meteorology, the Pennsylvania State University, University Park, PA 16802, United States;3. Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States;1. Institute e-Austria Timi?oara, Romania;2. Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA;1. 4-13-11 Hachi-Hon-Matsu-Minami, Higashi-Hiroshima 739-0144, Japan;2. Faculty of Education, Oita University, Dannoharu Oita-city 870-1192, Japan;3. Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan |
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| Abstract: | In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces. |
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