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# 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 $f(x_{1})\neq f(x_{2})$, 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 $f(x_{1})\neq f(x_{2})$, 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.

## Mathematics Subject Classification

26A06*no label found*70F25

*no label found*17B50

*no label found*81-00

*no label found*

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