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For every integer \( \mathrm{n} \), let \( \mathrm{a}_{\mathrm{n}} \) and \( \mathrm{b}_{\mathrm{n}} \) be real numbers. Let function \( f: \mathbb{R} \rightarrow \mathbb{R} \) be given by \( f(x)=\left\{\begin{array}{lll}a_{n}+\sin \pi x, & \text { for } & x \in[2 n, 2 n+1] \\ b_{n}+\cos \pi x, & \text { for } & x \in(2 n-1,2 n)\end{array}\right. \), for all integers \( n \).
\( \mathrm{P} \)
W
If \( f \) is continuous, then which of the following holds(s) for all \( \mathrm{n} \) ?
(A) \( a_{n-1}-b_{n-1}=0 \)
(B) \( a_{n}-b_{n}=1 \)
(C) \( a_{n}-b_{n+1}=1 \)
(D) \( a_{n-1}-b_{n}=-1 \)
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