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[parent] proof of generalized intermediate value theorem (Proof)
Theorem   Let $ f:X\rightarrow Y$ be a continuous map with $ X$ a connected space and $ Y$ a linearly ordered set in the order topology. If $ a,b\in X$ and $ y\in Y$ lies between $ f(a)$ and $ f(b)$, then there exists $ x\in X$ such that $ f(x)=y$.
Proof. Let $ a,b,y$ be as in the statement of the theorem. 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); furthermore, they are both non-empty, as $ f(a)$ is contained in one and $ f(b)$ is contained in the other. If $ y\notin f(X)$, then $ U\cup V$ constitutes a separation of the space $ f(X)$, contrary to the result that the continuous image of a connected space is connected. Thus there must exist $ x\in X$ such that $ f(x)=y$. $ \qedsymbol$
This version of the intermediate value theorem reduces to the familiar one from real analysis when $ X$ is taken to be a closed interval in $ \mathbb{R}$ and $ Y$ is taken to be $ \mathbb{R}$.

Bibliography

1
J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.



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See Also: order topology, total order, continuous, connected space, connectedness is preserved under a continuous map

Keywords:  continuous, connected, order, order topology

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Cross-references: closed interval, intermediate value theorem, image, contained, subspace topology, open subsets, disjoint, order topology, linearly ordered set, connected space, continuous map
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This is version 3 of proof of generalized intermediate value theorem, born on 2007-06-22, modified 2007-06-22.
Object id is 9639, canonical name is ProofOfGeneralizedIntermediateValueTheorem.
Accessed 903 times total.

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