08wHW3 - f which is improperly integrable on[0 ∞ and L =...

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MAT 125B Homework 3 M.Fukuda Please submit your answers at the discussion session on February 5 Tuesday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. i) Let I = (0 , 1] and f ( x ) = 1 /x p where p R . Find all p for which f is improperly intergrable in I . ii) By using i) show that the product of two improperly integrable functions is not neces- sarily improperly intergrable. iii) Find two functions where each function is not improperly integrable but the sum is improperly integrable. 2. i) Suppose we have a function
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Unformatted text preview: f which is improperly integrable on [0 , + ∞ ) and L = lim x → + ∞ f ( x ) exists. Then, show that L = 0. ii) Let f ( x ) = 1 n ≤ x < n + 2 − n , n ∈ N otherwise . Then, show that f is improperly integrable on [0 , + ∞ ) but lim x →∞ f ( x ) does not exists. 3. i) Let d ( x, y ) = b x − y b for x, y ∈ R . Show that d ( , ) satis±es the conditions of distance. ii) Let T : R n → R m and b b be the Euclidean norm de±nd in R n and R n . Prove that sup b x b =1 b T ( x ) b = sup b x bn =0 b T ( x ) b b x b . 1...
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