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 such that $f(x_1)\ne f(x_2)$, 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 function on a connected topological space $X$, and $x_1,x_2\in X$ with $f(x_1)\ne f(x_2)$, 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 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 (http://planetmath.org/ProofOfGeneralizedIntermediateValueTheorem) 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