A Common Generalization of the Intermediate Value Theorem and Rouché's Theorem

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.