Connectedness of the sphere

Question

There is a theorem in "Lecture notes on elementary Topology and Geometry" by Singer and Thorpe, that states:

Theorem: Let $S,T$ be topological spaces and $f:S \rightarrow T$; be continuous and surjective. If S is connected, then T is connected.

Now, given that $A=\mathbb{R^{n+1}}$\ {$0$} is connected, I can define $f:A \rightarrow S^n$ (where $S^n$ is the unit sphere) as follows:

$f(x)=f(x_{1},...x_{n+1})=(\dfrac {x_{1}}{\| x\|},...,\dfrac {x_{n+1}}{\| x\|})$.

So all I need to do is prove that $f$ is both continuous and surjective, and I'm having trouble doing both of those things. Can anybody give me any ideas?

Thanks in advance

Answer 1

Surjectivity: every point $x$ of $S^n$ is also in $A$, and you have $f(x)=x$, beacuse $\lVert x \rVert=1$.

Continuity: Each component function of $f$ is continuous beacuse it is quotient of two continuous functions: a projection and the norm funcion, and the norm function never vanishes in $A$.