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Unformatted text preview: 18.100B, Fall 2002, Homework 8 solutions Was due by Noon, Tuesday November 19 Rudin: (1) Chapter 6, Problem 5 Solution. 1. No, it is not true that a bounded function, f on [ a, b ] with f 2 ∈ R ( α ) is necessarily in R ( α ) itself. We need a counterexample to see this. Take the function f = 1 at rational points and f = − 1 at irrational points. This is not integrable by the preceeding question (the difference between upper and lower sums is always 2( b − a )) . On the other hand f 2 = 1 , 2. If f is real-valued and bounded and f 3 ∈ R ( α ) then f ∈ R ( α ) as follows from Theorem 6.11 with φ ( t ) = t 1 / 3 the unique real cube root. (2) Chapter 6, Problem 7 Solution. (a) If f ∈ R on [0 , 1] then 1 1 c f ( x ) dx = f ( x ) dx − f ( x ) dx c 1 and if | f | ≤ M then | c f ( x ) dx | ≤ 2 Mc so 1 f ( x ) dx −→ f ( x ) dx as c c ↓ . [It is enough to say that c 1 f ( x ) dx depends continuously on c by Theorem 6.20....
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This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.
- Fall '09