# local homeomorphisms between real numbers

Let $I$ be an open interval and $f:I\to\mathbb{R}$ be a continuous map  . Then $f$ is a local homeomorphism if and only if $f$ is a homeomorphism onto image.

Proof. ,,$\Leftarrow$” If $f$ is a homeomorphism onto image, then (in particular) $f$ is monotonic and continuous  , thus $f(I)$ is open in $\mathbb{R}$ (please, see this entry (http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism) for more details). It is easy to see that therefore $f$ is a local homeomorphism.

,,$\Rightarrow$” Assume that $f$ is not a homeomorphism onto image. It is well known, that this implies that $f$ is not injective  (please, see this entry (http://planetmath.org/InjectiveMapBetweenRealNumbersIsAHomeomorphism) for more details). Let $x,y\in I$ be such that $x and $f(x)=f(y)$. Then there exists $c\in I$ such that $x and $c$ is a local maximum  of $f$. Thus (since $f$ is a Darboux function) for any $\varepsilon>0$ there are points $x_{\varepsilon},y_{\varepsilon}\in(c-\varepsilon,c+\varepsilon)$ such that $f(x_{\varepsilon})=f(y_{\varepsilon})$. This obviously implies that $f$ cannot be locally inverted around $c$. $\square$

Title local homeomorphisms between real numbers LocalHomeomorphismsBetweenRealNumbers 2013-03-22 18:53:50 2013-03-22 18:53:50 joking (16130) joking (16130) 5 joking (16130) Theorem msc 54C05