# generalized intermediate value theorem

###### Theorem.

Let $f:X\rightarrow Y$ be a continuous function   with $X$ a connected space and $Y$ a totally ordered set  in the order topology. If $x_{1},x_{2}\in X$ and $y\in Y$ lies between $f(x_{1})$ and $f(x_{2})$, then there exists $x\in X$ such that $f(x)=y$.

###### Proof.

The sets $U=f(X)\cap(-\infty,y)$ and $V=f(X)\cap(y,\infty)$ are disjoint open subsets of $f(X)$ in the subspace topology, and they are both non-empty, as $f(x_{1})$ is contained in one and $f(x_{2})$ is contained in the other. If $y\notin f(X)$, then $U\cup V$ constitutes a of the space $f(X)$, contradicting the hypothesis  that $f(X)$ is the continuous image of the connected space $X$. Thus there must exist $x\in X$ such that $f(x)=y$. ∎

This version of the intermediate value theorem reduces to the familiar one of http://planetmath.org/node/7599real analysis when $X$ is taken to be a closed interval  in $\mathbb{R}$ and $Y$ is taken to be $\mathbb{R}$.

## References

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Title generalized intermediate value theorem GeneralizedIntermediateValueTheorem 2013-03-22 17:17:44 2013-03-22 17:17:44 azdbacks4234 (14155) azdbacks4234 (14155) 8 azdbacks4234 (14155) Theorem msc 26A06 OrderTopology TotalOrder Continuous ConnectedSpace ConnectednessIsPreservedUnderAContinuousMap