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Math 447, Homework 2.∗ 1. Let f be a bounded function, and α increasing. (a) Show that if f ∈ Rα [a, b], then f 2 ∈ Rα [a, b] (cf. Exercise 4). (b) Give an example where f 2 ∈ Rα [a, b] but f 6∈ Rα [a, b] (cf. Exercise 5). 2. Exercise 14.30 (page 226). Remember that a step function has only finitely many steps. If you are going to use Exercise 7, you of course need to solve it first. 3. Exercise 14.31 (page 226). 4. Exercise 14.37 (page 231). Quiz 2. See page 239 for the definition of the norm integral, for general bounded functions f and α. (a) Exercise 14.57 (page 240). (b) State and prove a version of Theorem 14.10 for the norm integral. ∗ c 2016 by Michael Anshelevich. 1 Math 446, Honors Homework 2 1. Let f be bounded a function, and α increasing. (a) Show that if f ∈ Rα [a, b], then f 2 ∈ Rα [a, b] (cf. Exercise 4). (b) Give an example where f 2 ∈ Rα [a, b] but f 6∈ Rα [a, b] (cf. Exercise 5). 2. Exercise 14.30 (page 226). Remember that a step function has only finitely many steps. If you are going to use Exercise 7, you of course need to solve it first. 3. Exercise 14.32 (page 226). One way to do this is to prove the contrapositive. How do you show that a function is not Riemann-Stieltjes integrable? 4. Exercise 14.37 (page 231). Honors Quiz 2. See page 239 for the definition of the norm integral, for general bounded functions f and α. (a) Exercise 14.57 (page 240). (b) State and prove a version of Theorem 14.10 for the norm integral.