For every integer n, let $a_n$ and $b_n$ be real numbers. 
Let $f:R\to R$ be given by
$f\left(x\right)=\{\begin{array}{ccc}{a}_n+\sin \pi x& ,for& x\in [2n,2n+1]\\ b_n+\cos \pi x& ,for& x\in (2n-1,2n)\end{array},$ for all integers n.
If f is continuous, then which of the following does not hold for all n.