Is there a single function describing this piecewise function?

Question

I am examining the piecewise function given by the following. $f(x) = \pi - \arctan(\frac{2x}{1-x^2})$ when $0 \leq x < 1$, $f(x) = \frac{\pi}{2}$ when $x=1$, and $f(x) = \arctan(\frac{2x}{x^2-1})$ when $x > 1$. One can verify that this defines a continuous curve. My question is, instead of having to use all these to describe this curve, is there possibly a simpler way to express it? Hopefully using one function?

Answer 1

It is straightforward to see that $f(x)=\pi-2\arctan(x)$ for all $x\geq 0$.

Indeed, let $t=\arctan(x)$ so that $0\leq t<\pi/2$ and $x=\tan(t)$. Then $$\frac{2x}{1-x^2}=\frac{2\tan(t)}{1-\tan^2(t)}=\tan(2t),$$ hence $$\arctan\left(\frac{2x}{1-x^2}\right)= \begin{cases} 2t,& 0\leq x<1;\\ 2t-\pi,& x>1. \end{cases}$$