Finding a bijective function from $\mathbb R \setminus \{0\}$ to $\mathbb R$ Basically I wanna show that $|\mathbb R \setminus \{0\}|=|\mathbb R|$, so I need to find a bijective function $f: \mathbb R \setminus \{0\} \rightarrow \mathbb R$.

What I tried: 

$\mathbb R \setminus \{0\} = (-\infty,0) \cup (0, \infty)$

And playing with that. I showed $\mathbb R \setminus \{0\}  \sim (-1,0) \cup (0,1) $. What can I do to continue? 

Thanks!
Try this: $$f(x)=\begin{cases} x &amp; x\notin \mathbb{N}\\ x+1 &amp; x\in \mathbb{N}\end{cases}$$

Note: for this function, the domain is $\mathbb{R}$, and $\mathbb{N}=\{0,1,2,3,\ldots\}$.  This function is the inverse of the one sought by OP.
Each has the same cardinality. Thus, each is in bijection with the corresponding cardinal number. The composition of bijection is a bijection, so the composition bijects your two sets.