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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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gk@gmail.com

Posts: 134
Registered: 11/12/12
Continuous, locally 1-1 function from Reals to Reals is globally 1-1.
Posted: Apr 19, 2013 3:03 PM
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Hi, all:

I have been able to show that a continuous, locally 1-1 function from

Reals to Reals is globally 1-1. by locally 1-1 I mean every point x in R

has a 'hood ( neighborhood) U_x where f|_U_x is 1-1. But I would like to

know if locally 1-1 enough is alone, without continuity ( I know f being

locally monotonic is enough).

For the continuous case, we just show f must be monotonic in every 'hood

where it is 1-1 . Then we assume there are x,y with f(x)=f(y) . But then

there is a chain of monotonic 'hoods joining x to y, forcing all these chains

to be either all increasing or all decreasing ( we encase x,y in [-M,M]

so that there is a finite cover by locally 1-1 'hoods).

Is this true if f is just locally 1-1 but not continuous?

Thanks.



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