An $1-1$ function

Does there exist a function $f:\mathbb{R} \rightarrow \mathbb{R}$ with $f(\mathbb{R})=\mathbb{R}^*$ that is $1-1$ ?

Solution

Of course. There are plenty of examples for which the above holds. For example one such function would be $f(x)=x$ unless $x$ is natural. In naturals we define $f(0)=1, \; f(1)=2, \; f(2)=3$ etc. Another function would be:

$$f(x) = \begin{cases}
      x &,&x <0  \\
      1-x &, &  0< x \leq 1  \\
      3-x &, &1< x \leq 2  \\
      ...\\
      2n+1-x &, & n< x \leq n+1\\
      ...\\
   \end{cases}$$

Of course such function cannot be continuous. We can see that from Bolzano. If $f$ takes the values $\pm 1$ then it shall also take the value $0$.