09hw1 - the L 1 metric. 5. Let < X,ρ > be a compact...

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Mathematics 241, Problem Set #1, due August 31, 2009 Write clear proofs that have good grammar and consist of complete sentences. 1. Prove that a function from a Metric Space < X,ρ > to a metric space < Y,σ > is contin- uous at x±X if and only if, given and ε > 0, there exists a δ > 0 such that σ ( f ( y ) ,f ( x )) ε whenever y satisfies ρ ( y,x ) δ. 2. Royden, Page 147, #12. 3. Royden, Page 147, #17(a). 4. In class an example was sketched that shows that C [ a,b ] is not complete in the L 1 metric. Prove that the sequence of continuous functions described is indeed a Cauchy sequence in
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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 inte-grable: 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.

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