intermediate value theorem

If $f$ is a real-valued continuous function on the interval $[a,b]$ ,and $x_{1}$ and $x_{2}$ are points with $a\\leq x_{1}<x_{2}\\leq b$ such that $f(x_{1})\eq f(x_{2})$ ,then for every $y$ strictly between $f(x_{1})$ and $f(x_{2})$ there is a $c\\in(x_{1},x_{2})$ 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 connected topological space $X$ ,and $x_{1},x_{2}\\in X$ with $f(x_{1})\eq f(x_{2})$ ,then for every $y$ between $f(x_{1})$ and $f(x_{2})$ there is a $\\xi\\in X$ such that $f(\\xi)=y$ .(However, this “generalization” 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.

单词	IntermediateValueTheorem
释义	intermediate value theorem If $f$ is a real-valued continuous function on the interval $[a,b]$ ,and $x_{1}$ and $x_{2}$ are points with $a\\leq x_{1}<x_{2}\\leq b$ such that $f(x_{1})\eq f(x_{2})$ ,then for every $y$ strictly between $f(x_{1})$ and $f(x_{2})$ there is a $c\\in(x_{1},x_{2})$ 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 connected topological space $X$ ,and $x_{1},x_{2}\\in X$ with $f(x_{1})\eq f(x_{2})$ ,then for every $y$ between $f(x_{1})$ and $f(x_{2})$ there is a $\\xi\\in X$ such that $f(\\xi)=y$ .(However, this “generalization” 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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