Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant $a$, the equation $f\left(x+a\right)=\frac{1}{2}+\sqrt{f\left(x\right)-{\left(f\left(x\right)\right)}^2}$ holds for all $x$. Then $f\left(x\right)$ is periodic function with period equal to

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 $n$?