On Friday, April 19, 2013 3:03:41 PM UTC-4, hbe...@gmail.com wrote: > 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.