The Continuity Postulate in Economic Theory: A Deconstruction and an Integration

TL;DR

This paper systematically deconstructs and integrates the continuity postulate in economic theory, revealing multiple meanings and equivalences under various mathematical structures, and exploring their behavioral implications.

Contribution

It offers a novel systematic analysis of the multiple meanings of continuity and their equivalences in economic choice theory, extending foundational work by key theorists.

Findings

Multiple meanings of continuity are equivalent under certain mathematical structures.

Restricted and full continuity have distinct but related behavioral implications.

New theorems clarify the role of continuity in economic decision models.

Abstract

This paper presents six theorems and ten propositions that can be read as deconstructing and integrating the continuity postulate under the rubric of pioneering work of Eilenberg, Wold, von Neumann-Morgenstern, Herstein-Milnor and Debreu. Its point of departure is the fact that the adjective continuous applied to a function or a binary relation does not acknowledge the many meanings that can be given to the concept it names, and that under a variety of technical mathematical structures, its many meanings can be whittled down to novel and unexpected equivalences that have been missed in the theory of choice. Specifically, it provides a systematic investigation of the two-way relation between restricted and full continuity of a function and a binary relation that, under convex, monotonic and differentiable structures, draws out the behavioral implications of the postulate.