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Math Help - Real analysis questions - thanks alot!

  1. #1
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    Real analysis questions - thanks alot!

    Hey guys,
    i'm working on real analysis and i'm stuck on these 2 questions:
    1) Prove that the annulus A = {z belongs to R2| r <=|z|<=R} (where R>r>0) is connected.

    2) f: R -> R is differentiable
    Suppose there exists L < 1 such that f'(x) < L for all R
    (a) Prove that f has a unique fixed point
    (b) Show by example that (a) fails if L = 1
    Thanks alot guys!
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    Quote Originally Posted by pc31 View Post
    2) f: R -> R is differentiable
    Suppose there exists L < 1 such that f'(x) < L for all R
    (a) Prove that f has a unique fixed point
    (b) Show by example that (a) fails if L = 1
    2(a)
    Let g(x) = x-f(x). Then g'(x) = 1-f'(x)>0. Hence g is a strictly increasing function. A strictly increasing function can only cross the x-axis at most once.

    Suppose g(x) doesn’t cross the x-axis at all. The only way this is possible for a strictly increasing differentiable (and thus continuous) function is if either (i) g(x) is negative for all x and g'(x)\to0 as x\to\infty or (ii) g(x) is positive for all x and g'(x)\to0 as x\to-\infty. In either case, we have that f'(x)\to1 as x\to\infty. This contradicts the fact that f'(x) is bounded above by L<1. (Get it? \lim_{x\to\infty}{f'(x)}=1 means that f'(x) gets arbitrarily close to 1 for sufficiently large values of x; in particular, it can get within 1-L of 1, in which case, we would have the contradiction that f'(x)>L for these sufficiently large values of x.)

    Hence g must cross the horizontal axis exactly once; in other words, f has a unique fixed point.

    2(b)
    For this, just choose any strictly increasing function g(x)<0 such that \lim_{x\to\infty}{g'(x)}=0 (or g(x)>0 such that \lim_{x\to-\infty}{g'(x)}=0). For example, g(x)=-\mathrm{e}^{-x}, i.e. f(x)=x+\mathrm{e}^{-x}.
    Last edited by algebraic topology; June 30th 2008 at 04:26 PM.
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    Quote Originally Posted by pc31 View Post
    Hey guys,
    i'm working on real analysis and i'm stuck on these 2 questions:
    1) Prove that the annulus A = {z belongs to R2| r <=|z|<=R} (where R>r>0) is connected.
    Hint: It is sufficient to sohw that the annulus is pathwise connected.
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    Quote Originally Posted by ThePerfectHacker View Post
    Hint: It is sufficient to sohw that the annulus is pathwise connected.

    Can you please specify? Thank you!
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    Quote Originally Posted by pc31 View Post
    Can you please specify? Thank you!
    A set S is pathwise connected iff between any two distinct points there exists a continous function (path) joining those two point and lying wholly within the set. If S is pathwise connected then it is connected. (The converse is not true).
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  6. #6
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    Quote Originally Posted by pc31 View Post
    1) Prove that the annulus A = {z belongs to R2| r <=|z|<=R} (where R>r>0) is connected.
    1
    To prove path-connectedness …

    Let (r_1,\theta_1) and (r_2,\theta_2), where r\leq r_1,r_2\leq R, be any two points in A, expressed in polar co-ordinates. Define f:[0,1]\to \mathbb{R}^2 by

    f(t)=\left((1-t)r_1+tr_2,(1-t)\theta_1+t\theta_2\right)

    Note that f(0)=(r_1,\theta_1) and f(1)=(r_2,\theta_2). Now all you need to do is to show that for any t\in[0,1], r\leq(1-t)r_1+tr_2\leq R. Then f would be a path in A connecting (r_1,\theta_1) and (r_2,\theta_2); hence any two points in A can be connected by a path in A and so A is path-connected.
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