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Topic: Continuous, locally 1-1 function from Reals to Reals is globally 1-1.
Replies: 10   Last Post: Apr 20, 2013 2:57 PM

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 scattered Posts: 92 Registered: 6/21/12
Re: Continuous, locally 1-1 function from Reals to Reals is globally 1-1.
Posted: Apr 19, 2013 4:31 PM
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On Friday, April 19, 2013 3:03:41 PM UTC-4, hbe...@gmail.com wrote:
> Hi, all:
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> I have been able to show that a continuous, locally 1-1 function from
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> Reals to Reals is globally 1-1. by locally 1-1 I mean every point x in R
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> has a 'hood ( neighborhood) U_x where f|_U_x is 1-1. But I would like to
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> know if locally 1-1 enough is alone, without continuity ( I know f being
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> locally monotonic is enough).
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> For the continuous case, we just show f must be monotonic in every 'hood
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> where it is 1-1 . Then we assume there are x,y with f(x)=f(y) . But then
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> there is a chain of monotonic 'hoods joining x to y, forcing all these chains
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> to be either all increasing or all decreasing ( we encase x,y in [-M,M]
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> so that there is a finite cover by locally 1-1 'hoods).
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> Is this true if f is just locally 1-1 but not continuous?
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> Thanks.

Consider f(x) = x - floor(x)

Date Subject Author
4/19/13 gk@gmail.com
4/19/13 David C. Ullrich
4/19/13 gk@gmail.com
4/20/13 David C. Ullrich
4/20/13 fom
4/20/13 gk@gmail.com
4/19/13 scattered
4/19/13 quasi
4/19/13 fom
4/20/13 gk@gmail.com
4/20/13 fom

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