Pseudocontinuity is Necessary and Sufficient for Order-preserving Continuous Representations

Abstract

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.