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Re: Continuous, locally 11 function from Reals to Reals is globally 11.
Posted:
Apr 19, 2013 4:31 PM


On Friday, April 19, 2013 3:03:41 PM UTC4, hbe...@gmail.com wrote: > 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.
Consider f(x) = x  floor(x)



