# 05 - ASSIGNMENT 5 Â SOLUTIONS MAT 572 A Â FALL 2007 Problem...

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Unformatted text preview: ASSIGNMENT 5 Â· SOLUTIONS MAT 572 A Â· FALL 2007 Problem 1 ( cf. [1, Exercise IV.1 #7]) . Show that Î³ : [0 , 1] â†’ C defined by Î³ ( t ) = t + i t sin(1 /t ) t 6 = 0 t = 0 is a path, but is not rectifiable. Proof. Since lim t â†’ + t sin(1 /t ) = 0, Î³ is continuous on [0 , 1], so Î³ is a path. See Figure 1 for the sketch. For the rest, consider partitions of the form P n = , 2 (2 n- 1) Ï€ , . . . , 2 5 Ï€ , 2 3 Ï€ , 2 Ï€ , 1 . (There are n + 2 elements of P n , which we label as usual as t = 0 < t 1 < Â·Â·Â· < t n +1 = 1.) Then for 2 â‰¤ k â‰¤ n , we have sin(1 /t k ) = (- 1) n- k , so the successive differences become sums: v ( Î³ ; P n ) = n +1 X k =1 | Î³ ( t k )- Î³ ( t k- 1 ) | â‰¥ n X k =2 | Im Î³ ( t k )- Im Î³ ( t k- 1 ) | = n X k =2 | t k sin(1 /t k )- t k- 1 sin(1 /t k- 1 ) | = n X k =2 ( t k- 1 + t k ) = 2 Ï€ 1 2 n- 1 + 1 2 n- 3 + Â·Â·Â· + 1 5 + 1 3 + 1 3 + 1 1 â‰¥ 2 Ï€ 1 2 n- 1 + 1 2 n- 2 + Â·Â·Â· + 1 5 + 1 4 + 1 3 + 1 2 = 2 n- 1 X k =2 1 k ....
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05 - ASSIGNMENT 5 Â SOLUTIONS MAT 572 A Â FALL 2007 Problem...

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