The Intermediate Value Theorem and Decision-Making in Psychology and Economics: An Expositional Consolidation

Abstract

On taking the intermediate value theorem (IVT) and its converse as a point of departure, this paper connects the intermediate value property (IVP) to the continuity postulate typically assumed in mathematical economics, and to the solvability axiom typically assumed in mathematical psychology. This connection takes the form of four portmanteau theorems, two for functions and the other two for binary relations, that give a synthetic and novel overview of the subject. In supplementation, the paper also surveys the antecedent literature both on the IVT itself, as well as its applications in economic and decision theory. The work underscores how a humble theorem, when viewed in a broad historical frame, bears the weight of many far-reaching consequences; and testifies to a point of view that the apparently complicated can sometimes be under-girded by a most basic and simple execution.

On Separate Continuity and Separate Convexity: A Synthetic Treatment for Functions and Sets

“…REMARK 4.5. Different continuity concepts of a binary relation or a correspondence, including the properties of straight lines and curves, have been extensively used in economics and psychology; see for example [2,10,11,13,23,24]. We leave it for future research to study the extensions of the results in this paper to infinite-dimensional spaces, and their implications to the continuity of correspondences and binary relations.…”

“…A straight line in X ⊆ R n is defined as the intersection of a one-dimensional subset of the affine hull of X. Next, we introduce a topological property that is motivated by separate continuity of a function that imposes continuity restricted to straight lines parallel to a coordinate axis; see [9,16,25] for classic results and [4,10] for recent surveys on different continuity postulates.…”

“…Theorem 3.1 contributes to this literature by studying the relationship between separate and joint continuity of functions under convexity, or concavity, an assumption that is weaker than monotonicity. See also [3,14,17] for relationships between separate and joint continuity in infinite-dimensional spaces, and [4,10,18] for surveys. Moreover, lower semicontinuity under concavity and upper semicontinuity under convexity of functions is studied in [6,8].…”

“…If a function defined on is piecewise monotone in (or monotone in ), then the arguments in [22, page 156] imply that separate continuity is equivalent to the following intermediate value property in the variable : for all and all , , and all between and , there exists such that . See [10] for an investigation of the intermediate value theorem in two applied registers.…”