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intermediate value theorem (Theorem)

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.) 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.



"intermediate value theorem" is owned by yark. [ full author list (3) | owner history (2) ]
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See Also: Rolle's theorem, mean-value theorem, continuous


Attachments:
proof of intermediate value theorem (Proof) by yark
intermediate value theorem for extended real numbers (Theorem) by matte
Bolzano's theorem (Theorem) by pahio
proof of generalized intermediate value theorem (Proof) by azdbacks4234
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Cross-references: order topology, codomain, topological space, connected, Bolzano's theorem, strictly, points, interval, continuous function
There are 18 references to this entry.

This is version 10 of intermediate value theorem, born on 2001-10-20, modified 2007-08-02.
Object id is 423, canonical name is IntermediateValueTheorem.
Accessed 14972 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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