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How do you prove that $ f(x,y) = x^y $ at $ [0,1]\times[1,\infty] $ I tried to prove this by definition, but got stuck when trying to show that the distance between two points approaches $\epsilon$ when they are relatively close.

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Google Brouwer's fixed point theorem, it should be easy to proceed from there.

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The function in question is from a subset of $\mathbb{R}^2$ to $\mathbb{R}$, so the notion of a "fixed point" doesn't exist. Moreover, the domain, $[0,1] \times [1,\infty]$ is not compact, and even if it were, the fixed point theorem requires continuity as an assumption, not as a conclusion. Your answer doesn't make any sense. –  Goos May 14 '13 at 7:42
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