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'intermediate value theorem'
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| Title of object: |
intermediate value theorem |
| Canonical Name: |
IntermediateValueTheorem |
| Type: |
Theorem |
| Created on: |
2001-10-20 22:26:16 |
| Modified on: |
2007-06-15 07:09:04 |
| Classification: |
msc:26A06 |
Revision comment (for changes between this and next version):
Preamble:
Content:
\PMlinkescapeword{order}
If $f$ is a real-valued continuous function on the interval $[a,b]$,
and $x_1$ and $x_2$ are points with $a\le x_1<x_2\le b$
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.
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