SCHWEITZER 1995 Problem 3

Denote $\langle x\rangle$ the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers $a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n$ such that $$f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle$$ for every x iff there is a k such that $$\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)$$ is constant.