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Some problems on mountain climbing Hung, Patrick Chia-Ling

Abstract

Let f and g be two continuous, real-valued functions defined on [0,1] with f(0) = g(0) and f(l) = g(l). The main result of this thesis is to characterize the property that (0,0) and (1,1) are in the same connected component of G(f,g) = {(x,y)|f(x)=g(y)}. In Chapter I, we study conditions implying that (0,0) and (1,1) are in the same connected component of G(f,g), where f and. g are not necessarily real-valued functions. We obtain theorems to characterize [0,1], In Chapter II, we give a simple proof of a theorem by Sikorski and Zarankiewicz. In Chapter III, we obtain our main result. In Chapter IV, we study pathwise connectedness in G(f,g) and give some applications. In Chapter V, we study the question of sliding a chord of given length along a path. An example is given to show that this is not always possible.

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