Let $\phi :\mathbb{R}^{n+1}\setminus\{0\}→\mathbb{R}_+×S^n$ bedefined as $$\phi(x)=\phi(x_1,\cdots,x_{n+1})=\left(\|x\|,\frac{x_1}{\|x\|},\cdots,\frac{x_{n+1}}{\|x\|}\right).$$
I have to prove that $\phi$ is continuous, surjective and injective.
For the injective part I assumed that if $x\neq x'$ then $x_{i}\neq x'_{i}$ for at least some $i$, then if $\|x\|\neq \|x'\|$ we are done, however if $\|x\|= \|x'\|$ then $\dfrac{x_i}{\|x\|}\neq \dfrac{x'_i}{\|x\|}$ for at least some $i$. Is this correct?
For the surjective part I do not know what to do.
Thanks in advance for your help.
Here $\phi (x)=(|x|,\frac{x}{|x|})$. If $ \phi (x)=\phi (y)$, then $|x|=|y|,\ \frac{x}{|x|}=\frac{y}{|y|}$ so that $x=y$. injective
If $x\in S^n$, then $\phi(x)=(1,x)$ and $\phi(lx)=(l,x)$ for $l>0$ Hence surjective.
If $|x-y|<\delta$, then assume that $|x|>|y|$
Note that $$ |x-y|\geq ||x|-|y||,\ |\frac{x}{|x|}|y|-y|$$ so that \begin{align*}|\phi(x)-\phi (y)| &=|(|x|,\frac{x}{|x|}) - ( |y|,\frac{y}{|y|} )|\\&\leq (|x|,0)-(|y|,0)| +|(0,\frac{x}{|x|}) - ( 0,\frac{y}{|y|} )|\\& \leq ||x|-|y|| + | \frac{x}{|x|} |y| -y| |y| \\&\leq \delta +\delta |y| \end{align*}
Hence it is continuous