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2008/2009 Pseudocontinuity is Necessary and Sufficient for Order-preserving Continuous Representations
Vincenzo Scalzo
Real Anal. Exchange 34(1): 239-248 (2008/2009).


Using some properties of pseudocontinuous functions - a recent generalization of continuous functions - and without invoking Debreu's Open Gap Theorem, we solve the following problem: given a pseudocontinuous function $v$, find a continuous function $u$ such that $u(x)>u(y)$ if and only if $v(x)>v(y)$. We show that this problem can be solved only for pseudocontinuous functions. Finally, we obtain a new proof on the existence of continuous numerical representations for continuous, transitive and total binary relations.


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Vincenzo Scalzo. "Pseudocontinuity is Necessary and Sufficient for Order-preserving Continuous Representations." Real Anal. Exchange 34 (1) 239 - 248, 2008/2009.


Published: 2008/2009
First available in Project Euclid: 19 May 2009

MathSciNet: MR2527137

Primary: 91B02
Secondary: 91B08

Keywords: continuous numerical representations of binary relations , order-preserving functions , pseudocontinuous functions

Rights: Copyright © 2008 Michigan State University Press

Vol.34 • No. 1 • 2008/2009
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