Is there a bijection from $[0,1]$ to $\mathbb{R}$? I'm looking for a bijection from the closed interval $[0,1]$ to the real line. I have already thought of $\tan(x-\frac{\pi}{2})$ and $-\cot\pi x$, but these functions aren't defined on 0 and 1.
Does anyone know how to find such a function and/or if it even exists?
Thanks in advance!

A continuous bijection can't exists because $[0,1]$ is a compact set and continuous functions send compacts in compacts. You can look for a non-continuous bijection, that exists because $[0,1]$ and $\mathbb R$ have the same cardinality. It follows from Cantor-Bernstein theorem, that states "given two sets $A, B$ if exist two injective functions $f:A\rightarrow B \ g:B\rightarrow A$" than exists a bijective function $h:A\rightarrow B$".
