Re: (un)stable fixed points

In article <[EMAIL PROTECTED]>,
David C. Ullrich  <[EMAIL PROTECTED]> wrote:
>On 3 Dec 2003 09:06:36 GMT, [EMAIL PROTECTED] (Robert Israel) wrote:

>>In article <[EMAIL PROTECTED]>,
>>billy d. <[EMAIL PROTECTED]> wrote:
>>>f:R-->R differentiable, continuous derivative, and f(a)=a for some a
>>>in R.

>>>if |f'(a)|<1, then the sequence x_n=f(x_n-1) converges to a when x_0
>>>is sufficiently close to a.

>>>if |f'(a)|>1, then there exists c_0>0 s.t. for all x_0=/=a,
>>>|x_N-a|>c_0 for some positive integer N.

>>>the first part was coming along nicely. it seems that i am close to
>>>showing that f is a contraction from [a-d,a+d] for some d>0.

>>Not true.  

>Counterexample? It seems true to me - I hesitate to post what
>seems like the easy proof since we're just giving hints.
>Possibly I'm being stupid again (or possibly you missed
>the fact that f is _continuously_ differentiable?)

Oops, yes of course I missed that the derivative is continuous (which is 
not necessary for the result).  

Robert Israel                                [EMAIL PROTECTED]
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2