Let F(x) := x cos(Tl/x) for x E (0, 1] and F(O) := O.It will be seen that f := F' E
Chapter 10, Problem 19(choose chapter or problem)
Let F(x) := x cos(Tl/x) for x E (0, 1] and F(O) := O.It will be seen that f := F' E 'R.*[O,1]but that its absolute value If I = IF'I ft 'R.*[O,1]. (Here f(O) := 0.)(a) Show that F' and IF'I are continuous on any interval [c, 1],0 < c < 1 and f ~.'R.*[O,1].(b) Ifak:= 2/(2k+ l)andbk:= l/kfork E N,thentheintervals[ak,bk]arenon-overlappingand 1/ k ~ I;kk Ifl.(c) Since the series L~I 1/ k diverges, then If I ft 'R.*[O,1].
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