Let <I>(x):= x Icos(Tl/x)I for x E (0, 1] and let <1>(0):= O.Then <I>is continuous on

Chapter 10, Problem 21

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Let (x):= x Icos(Tl/x)I for x E (0, 1] and let <1>(0):= O.Then is continuous on [0, 1] and'(x)exists for x ft E := {OJU {ak : kEN}, where ak := 2/(2k + 1). Let qJ(x) := '(x)forx ft E and qJ(x) := 0 for x E E. Show that qJis not bounded on [0, 1]. Using the FundamentalTheorem 10.1.9 with E countable, conclude that qJE 'R.*[O,1] and that I;qJ = (b) - (a)for a, bE [0, 1]. As in Exercise 19, show that IqJlft 'R.*[O,1].