For every integer

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

For every integer nn, let ana_n and bnb_n be real numbers. Let function f:R→Rf : R \to R be given by f(x)={an+sin⁡πx,for x∈[2n,2n+1]bn+cos⁡πx,for x∈(2n−1,2n]f(x) = \begin{cases} 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{cases} for all integers nn. If ff is continuous, then which of the following hold(s) for all nn? (1) an−1−bn−1=0a_{n-1} - b_{n-1} = 0 (2) an−bn=1a_n - b_n = 1 (3) an−bn+1=1a_n - b_{n+1} = 1 (4) an−1−bn=−1a_{n-1} - b_n = -1

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