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February 2006 A Common Generalization of the Intermediate Value Theorem and Rouché's Theorem
Richard Bayne, Terrence Edwards, Myung H. Kwack
Missouri J. Math. Sci. 18(1): 26-32 (February 2006). DOI: 10.35834/2006/1801026

Abstract

A simple proof of a theorem unifying Bolzano's Theorem [8], the Intermediate Value Theorem, Rouché's Theorem [3] and its extensions to differentiable maps to $\mathbb{R}^n$ [2, 6, 9] is obtained. This unifying theorem in particular shows that in Professor Baker's [1] examples where the number of solutions of $f(x) = y$ for a continuous map $f \colon B^2 \to \mathbb{R}^2$, $y \not \in f ( \partial B^2 )$, from the unit ball $B^2$ in the plane $\mathbb{R}^2$ is not exactly the absolute value of the winding number of the curve $f ( \partial B^2)$ about $y$, the number of the connected components of the solution set counted with multiplicity coincides with the winding number.

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Richard Bayne. Terrence Edwards. Myung H. Kwack. "A Common Generalization of the Intermediate Value Theorem and Rouché's Theorem." Missouri J. Math. Sci. 18 (1) 26 - 32, February 2006. https://doi.org/10.35834/2006/1801026

Information

Published: February 2006
First available in Project Euclid: 3 August 2019

zbMATH: 1134.30003
Digital Object Identifier: 10.35834/2006/1801026

Subjects:
Primary: 30C15
Secondary: 55M25

Rights: Copyright © 2006 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.18 • No. 1 • February 2006
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