generalized intermediate value theorem


Let f:XY be a continuous functionMathworldPlanetmathPlanetmath with X a connected space and Y a totally ordered setMathworldPlanetmath in the order topology. If x1,x2X and yY lies between f(x1) and f(x2), then there exists xX such that f(x)=y.


The sets U=f(X)(-,y) and V=f(X)(y,) are disjoint open subsets of f(X) in the subspace topology, and they are both non-empty, as f(x1) is contained in one and f(x2) is contained in the other. If yf(X), then UV constitutes a of the space f(X), contradicting the hypothesisMathworldPlanetmath that f(X) is the continuous image of the connected space X. Thus there must exist xX such that f(x)=y. ∎

This version of the intermediate value theorem reduces to the familiar one of analysis when X is taken to be a closed intervalMathworldPlanetmath in and Y is taken to be .


  • 1 J. Munkres, TopologyMathworldPlanetmath, 2nd ed. Prentice Hall, 1975.
Title generalized intermediate value theorem
Canonical name GeneralizedIntermediateValueTheorem
Date of creation 2013-03-22 17:17:44
Last modified on 2013-03-22 17:17:44
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 8
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 26A06
Related topic OrderTopology
Related topic TotalOrder
Related topic Continuous
Related topic ConnectedSpace
Related topic ConnectednessIsPreservedUnderAContinuousMap