For the function $f\left(x\right)=x {\cot }^{-1}x, x\ge 0$

Let $f\left(x\right)$ be a differentiable function and $f\left(\alpha \right)=f\left(\beta \right)=0(\alpha <\beta ),$ then in the interval $(\alpha , \beta )$

Note: Answer in the book is incorrect.

For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function $f:R\to R$ be given by

$f\left(x\right)=\left\{\begin{array}{cll}{a}_n+\sin \pi x,& \text{ for }& x\in [2n,2n+1]\\ b_n+\cos \pi x,& \text{ for }& x\in (2n-1,2n)\end{array}\right.$ for all integers $n$.  If $f$ is continuous, then which of the following hold(s) for all $n$?