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Unformatted text preview: (b) Z [ n,n +1]  f  0 as n 0. (c) n Z [ n,n +1]  f  0 as n 0. (d) lim inf n n Z [ n,n +1]  f  = 0 3. Let f L 2 (0 , ) (Lebesgue measure). Prove the following: (a) Z x f ( t ) dt x 1 / 2 k f k 2 for x > 0. (b) lim x x1 / 2 Z x f ( t ) dt = 0. 4. Dene f ( x,y ) = ( x4 / 3 sin( 1 xy ) if 0 < y < x < 1 otherwise Prove or disprove: Z 1 Z 1 f ( x,y ) dxdy = Z 1 Z 1 f ( x,y ) dydx . 1...
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 Spring '11
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