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Question: prove that f 0 1 r defined by...

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Prove that f : (0, 1] → R defined by f(x)s-1s continuous but not uniformly continuous. Then explain why this example is not a contradiction to Lemma 5.19 5-14.
5.19 Let f la, b] R be continuous. Then f is uniformly continuous. Proof. Suppose for a contradiction that f is not uniformly continuous. Then there is an ε > 0 such that for all δ > 0 there are u, u E [a,b] such that 11-ul < δ and If) 6. Therefore, for all k e N with we can find numbers ui IL E la, b] so that luvl and(u. By the Bolzano- -f( u)| > ε. Therefore, for all k E N with δk :- we can find numbers uk, Uk Weierstrass Theorem, the bounded sequence {uk i has a subsequence(14km 100=1 that converges to a t e la, bl. Because for all m E N the inequality ukn holds, we conclude that linn vk = minn u㎞ = t. But then, because f is continuous, km inef (%) = f(t) = mimo f (u ) , which means minn | f (%)-f (ukm1 = 0, m→00 a contradiction to lf (uk)-/(Uk)| ε for all natural numbers k.
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