Propose a function $f:(0,1) \to \mathbb R$ such that $f$ is continous on $(0,1)$ and $range(f)=\mathbb R$.
My thoughts :
I've proved that $(0,1)$ has the same cardinality as $\mathbb R$. But in that proof, i never proposed the function which takes something from $(0,1)$ and gives something in $\mathbb R$. I just proved that it exists. That's the problem ... I have no idea about how a function's range can span the whole $\mathbb R$.
Any solutions or even good hints would be nice from you.
Thanks in advance.