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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 R [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 :B → R, y 6∈ f(∂B), from the unit ball B in the plane R is not exactly the absolute value of the winding number of the curve f(∂B) about y, the number of the connected components of the solution set counted with multiplicity coincides with the winding number.

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