Injective function from unit circle

Question

Let $S$ denote the set of points on the unit circle centred at $(0,0)$. Does there exist an injective function $f : S \rightarrow S \setminus \{(1,0)\}$?

Answer 1

Let $v_1=(1,0), v_2=(\cos 1, \sin 1), v_3=(\cos \frac1 2, \sin \frac 1 2 ),v_4=(\cos \frac 1 3, \sin \frac 1 3 ),....$. Define $f(v)=v$ if $f \notin \{v_1,v_2,...\}$, $f(v_1)=v_2,f(v_2)=v_3,...$. This gives a bijection from $S$ onto $S\setminus \{(1,0)\}$.

Answer 2

Sure. Just take a injective sequence of points $(x_n)$ on the unit circle, setting $x_1=(1,0)$, and define $f:S^1\to S^1\setminus\{(1,0)\}$ by $f(x)=\begin{cases} x_{j+1}\,,x=x_j\\x\,,x\notin(x_n)\end{cases}$.

For instance, you could let $x_n=e^{2\pi i/n}$.