# Canonical utility functions and continuous preference extensions

*Igor Kopylov*

*Journal of Mathematical Economics*, 2016, vol. 67, issue C, pages 32-37

**Abstract:**
I define canonical utility functions via an explicit formula that inherits semicontinuity, continuity, Cauchy continuity, and uniform continuity from preferences. This construction is used to (i)show Rader’s and Debreu’s theorems in a fast and transparent way,(ii)refine these results for Cauchy and uniformly continuous preferences on a metric space X,(iii)extend such preferences from X to any larger metric domain Y⊃X, while preserving Cauchy and uniform continuity respectively. By contrast, regular continuity does not guarantee that continuous preference extensions should exist even in standard decision theoretic frameworks. For example, continuous preferences over simple lotteries or finite menus need not have continuous extensions to Borel distributions or compact menus respectively.

**Keywords:** Continuous preferences; Debreu Theorem; Rader Theorem; Cauchy continuity; Preference extensions; Uniform continuity (search for similar items in EconPapers)

**Date:** 2016

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**Persistent link:** http://EconPapers.repec.org/RePEc:eee:mateco:v:67:y:2016:i:c:p:32-37

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