intermediate value theorem

If f is a real-valued continuous functionMathworldPlanetmathPlanetmath on the interval [a,b], and x1 and x2 are points with ax1<x2b such that f(x1)f(x2), then for every y strictly between f(x1) and f(x2) there is a c(x1,x2) such that f(c)=y.

Bolzano’s theorem is a special case of this.

The theorem can be generalized as follows: If f is a real-valued continuous function on a connectedPlanetmathPlanetmath topological spaceMathworldPlanetmath X, and x1,x2X with f(x1)f(x2), then for every y between f(x1) and f(x2) there is a ξX such that f(ξ)=y. (However, this “generalizationPlanetmathPlanetmath” is essentially trivial, and in order to derive the intermediate value theorem from it one must first establish the less trivial fact that [a,b] is connnected.) This result remains true if the codomain is an arbitrary ordered set with its order topology; see the entry proof of generalized intermediate value theorem ( for a proof.

Title intermediate value theorem
Canonical name IntermediateValueTheorem
Date of creation 2013-03-22 11:51:29
Last modified on 2013-03-22 11:51:29
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Theorem
Classification msc 26A06
Classification msc 70F25
Classification msc 17B50
Classification msc 81-00
Related topic RollesTheorem
Related topic MeanValueTheorem
Related topic Continuous