Let P be a Cantor subset of [0,1] and let {(a n , b n )} be the sequence of inte

Question

Let P be a Cantor subset of [0,1] and let {(an, bn)} be the sequence of intervals complementary to P in (0,1).
(a) On each interval [an,bn] construct a differentiable function such that

fn(an) = fn(bn) = (f'n) + (an) = (f'n) - (bn) = 0,

Limsupx-> an+f'(x) = limsupx-> bn- f'n(x) = 1

Liminfx-> an+f'(x) = liminfx-> bn- f'n(x) = -1,

and |fn(x)| (x - an)2 (x - bn)2 and |f'n(x)| is bounded by 1 in each interval [an,bn].

(b) Let g be defined on [0,1] by

g(x) = { fn(x), if x € (an,bn), n= 1,2,..

= { 0, if x € P

Sket a picture of the graph of g.

(c) Prove that g is differentiable on [0,1]

(d) Prove that g'(x) = 0 for each x € P

(e) Prove that g' is discontinuous at every point of P

Thanks.

Explanation / Answer

http://books.google.co.in/books?id=vA9d57GxCKgC&pg=PA289&lpg=PA289&dq=Let+P+be+a+Cantor+subset+of+%5B0,1%5D+and+let+{(an,+bn)}+be+the+sequence+of+intervals+complementary+to+P+in+(0,1&source=bl&ots=KP1mKl6PTr&sig=G1szZHN9nFi_Z_gmVq--HkOZEQs&hl=en&sa=X&ei=1aM0T7bYCcO8rAelhdW6Dw&ved=0CCIQ6AEwAA#v=onepage&q=Let%20P%20be%20a%20Cantor%20subset%20of%20%5B0%2C1%5D%20and%20let%20%7B(an%2C%20bn)%7D%20be%20the%20sequence%20of%20intervals%20complementary%20to%20P%20in%20(0%2C1&f=false

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