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单词 IntermediateValueTheorem
释义

intermediate value theorem


If f is a real-valued continuous functionMathworldPlanetmathPlanetmath on the interval [a,b],and x1 and x2 are points with ax1<x2bsuch 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 functionon 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 itone must first establish the less trivial fact that [a,b] is connnected.)This result remains trueif the codomain is an arbitrary ordered set with its order topology;see the entryproof of generalized intermediate value theorem (http://planetmath.org/ProofOfGeneralizedIntermediateValueTheorem)for a proof.

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更新时间:2025/5/4 6:23:25