local homeomorphisms between real numbers
Proof. ,,” If is a homeomorphism onto image, then (in particular) is monotonic and continuous, thus is open in (please, see this entry (http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism) for more details). It is easy to see that therefore is a local homeomorphism.
,,” Assume that is not a homeomorphism onto image. It is well known, that this implies that is not injective (please, see this entry (http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism) for more details). Let be such that and . Then there exists such that and is a local maximum of . Thus (since is a Darboux function) for any there are points such that . This obviously implies that cannot be locally inverted around .
|Title||local homeomorphisms between real numbers|
|Date of creation||2013-03-22 18:53:50|
|Last modified on||2013-03-22 18:53:50|
|Last modified by||joking (16130)|