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.
Citation
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
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