Let I = [0,1] be the closed unit interval. Suppose f is a continuous mapping from I to I. Prove that for one x an element of I, f(x) = x.(adsbygoogle = window.adsbygoogle || []).push({});

Proof:

Since [0,1] is compact and f is continuous, f is uniformly continuous.

This is where I'm stuck. I'm wondering if I can use the fact that since max {|x-y| = 1} if |x-y| = 1 then f(x) - f(y) < max {episolin}. This of course only occurs when WLOG x=1 y=0.

Stuck as far as the rest of it goes.

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# Another Analysis question: continuity and compactness

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