Change of variables from intinite to bounded support.

Question

I may be missing something simple, but I am stuck. My question:

I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in (-\infty, \infty)$, but in order to discretize the state space I need $x$ to be bounded. Can anyone suggest a simple, invertible function such that $f(x): \mathbb{R} \rightarrow [0, 1]$, or any other bounded interval for that matter?

Thanks!

Answer 1

The arctangent function is a nice bijective map from $\mathbb{R}$ to $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. It also happens to be monotonic, which is sometimes a desirable property when trying to decide between other possible maps. If necessary, you can transform the range into any open interval you wish by scaling and shifting.