Homeomorphism Proof of $\mathbb{R}^2/\{0\}$ and $Z := \{(x, y, z) | x^2+y^2 = 1\}$ I want to show that $$\mathbb{R}^2/\{0\}$$ and $$Z := \{(x, y, z) | x^2+y^2 = 1\}$$ are homeomorph. Therefore $$f:\mathbb{R}^2/\{0\} \to Z $$ with $f(x, y) := (\frac{x}{r},\frac{y}{r}, ln(r))$   with    $r :=  \sqrt{x^2+y^2}$  
I already showed that it is bijective and $f^{-1} = (xe^z, ye^z)$  
Has anyone an idea how to show that this function is continuous?

I will extends a bit my comment. There is no need to use $\delta$-$ \varepsilon$ proof if you know the following : 


*

*A function $f : \mathbb R^n \to \mathbb R^m, u \mapsto (f_1(u), \dots, f_m(u))$ is continuous if and only if all the functions $f_i, i=1, \dots, m$ are continuous. 

*$\log$ is a continuous function, $r(x,y) = \sqrt{x^2 + y^2}$ is a continuous function, projections are continuous functions.

*Compositions of continuous functions are continuous.

*If $f$ is continuous at $a$, $g$ continuous and non-vanishing at $a$, then $f/g$ is well defined and continuous at $a$. 

