For every integer n, let $$a_n$$ and $$b_n$$ be real numbers. Let function $$f : IR \rightarrow IR$$ be given by
$$ f(x) = \begin{cases}a_n + \sin \pi x & for x \in [2n , 2n + 1]\\b_n + \cos \pi x & for x \in [2n - 1 , 2n]\end{cases}$$, for all integers n.
If f is continuous, then which of the following hold(s) for all n?

A

$$a_{n-1} - b_{n-1} = 0$$

B

$$a_{n} - b_{n} = 1$$

C

$$a_{n} - b_{n+1} = 1$$

D

$$a_{n-1} - b_{n} = -1$$