Find an elementary function of calculus that maps an interval one to one onto $\displaystyle \mathbb{R}$.
We can build that function geometrically and represent intuitive look of such function.
$\displaystyle \mathbb{R}$ is the $\displaystyle x$ axis. We curve $\displaystyle (0,1)$ to half o a circle which his diameter parallel to $\displaystyle x$.
We see every point $\displaystyle x\in (0,1)$ as a point on half circle.
Now let us define $\displaystyle f(x)$:
For every $\displaystyle x\in (0,1)$ we will pass straight line joining the center of the circle with point $\displaystyle x\in (0,1)$ (which is on the half circle) and continue to $\displaystyle x$ axis(which is the whole $\displaystyle \mathbb{R}$.
The intersection point which that line makes with $\displaystyle x$ axis is $\displaystyle f(x)$.
It easy to see that f is one-to-one from $\displaystyle (0,1)$ onto $\displaystyle \mathbb{R}$.