generalized intermediate value theorem
Theorem.
Let $f\mathrm{:}X\mathrm{\to}Y$ be a continuous function^{} with $X$ a connected space and $Y$ a totally ordered set^{} in the order topology. If ${x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{\in}X$ and $y\mathrm{\in}Y$ lies between $f\mathit{}\mathrm{(}{x}_{\mathrm{1}}\mathrm{)}$ and $f\mathit{}\mathrm{(}{x}_{\mathrm{2}}\mathrm{)}$, then there exists $x\mathrm{\in}X$ such that $f\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}y$.
Proof.
The sets $U=f(X)\cap (-\mathrm{\infty},y)$ and $V=f(X)\cap (y,\mathrm{\infty})$ 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
- 1 J. Munkres, Topology^{}, 2nd ed. Prentice Hall, 1975.
Title | generalized intermediate value theorem |
---|---|
Canonical name | GeneralizedIntermediateValueTheorem |
Date of creation | 2013-03-22 17:17:44 |
Last modified on | 2013-03-22 17:17:44 |
Owner | azdbacks4234 (14155) |
Last modified by | azdbacks4234 (14155) |
Numerical id | 8 |
Author | azdbacks4234 (14155) |
Entry type | Theorem |
Classification | msc 26A06 |
Related topic | OrderTopology |
Related topic | TotalOrder |
Related topic | Continuous |
Related topic | ConnectedSpace |
Related topic | ConnectednessIsPreservedUnderAContinuousMap |