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Unformatted text preview: the L 1 metric. 5. Let < X,ρ > be a compact metric space and suppose that f is a continuous function from X tp R . Prove that there exist x and x such that: f ( x ) = sup x±X f ( x ) , f ( x ) = inf x±X f ( x ) . 6. Prove directly from the deﬁnition that the following function on [0 , 1] is Riemann integrable: f ( x ) = 1 if 0 ≤ x < 1 2 . 17 if x = 1 2 . 2 if 1 2 < x ≤ 1 . (1) 7. Royden, Page 16, #18. 8. Royden, Page 39, #7. 9. Royden, Page 39, #17. 10. Royden, Page 46, #36....
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.
 Spring '09
 Reed
 Math

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