Continuous bijection from $(0,1)$ onto $\mathbb{R}$ has a continuous inverse

Question

Let $f$ be a continuous bijection from $(0,1)$ onto $\mathbb{R}$. Then show that $f^{-1}$ is continuous.

We can use the fact that $f$ has to be a monotone function and so is $f^{-1}$. Together with the fact that surjective real valued monotone function is continuous we can conclude. Is there any other approach?