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Continuous, locally 11 function from Reals to Reals is globally 11.
Posted:
Apr 19, 2013 3:03 PM


Hi, all:
I have been able to show that a continuous, locally 11 function from
Reals to Reals is globally 11. by locally 11 I mean every point x in R
has a 'hood ( neighborhood) U_x where f_U_x is 11. But I would like to
know if locally 11 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 11 . 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 11 'hoods).
Is this true if f is just locally 11 but not continuous?
Thanks.



