Let \II(x) := x21cos(Tl/x)I for x E (0, 1] and \11(0):= O.Then \II is continuous on [0

Chapter 10, Problem 22

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Let \II(x) := x21cos(Tl/x)I for x E (0, 1] and \11(0):= O.Then \II is continuous on [0, 1] and\f1'(x) exists for x ftEI:= {ak}. Let 1/1(x):= \II'(x) for x ft EI and 1/1(x):= 0 for x EEl'Show that 1/1is bounded on [0, 1] and (using Exercise 7.2.11) that 1/1E 'R.[O,1]. Show thatI: 1/1= \II(b) - \f1(a) for a, b E [0, 1]. Also show that 11/11E 'R.[O, 1].

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