Bijective continuous function between $(0 ,\infty)$ and $\mathbb{R}$. I've been working on some problems in topology, and it is odd to start
working again on sets, so I've needed some help. I want to build a
bijective continuous function $f$ such that its inverse is continuous
from $(0,\infty)$ and $(-\infty,\infty).$ I have a feeling that this
problem is similar to the Hilbert's Hotel problem, but I am having
issues figuring out the trick to make this surjective. I understand that
I need to negate the negatives, but how to differentiate the mapping of
the negatives and the positives to $\mathbb{R}^+$ seems peculiar to me.

Let $f:(0,\infty)\to(-\infty,\infty)$ be defined by:
$$f(x)=x-\frac1x$$
It's inverse $f^{-1}:(-\infty,\infty)\to(0,\infty)$ is defined by:
$$f^{-1}(x)=\frac x2+\sqrt{\frac{x^2}4+1}$$
The function $x\mapsto\ln x$ is simpler, in my opinion.
