Rochberg's coboundary theorem provides conditions under which the equation (I-T)y = x is solvable in y. Here T is a unilateral shift on Hilbert space, I is the identity operator and x is a given vector. The conditions are expressed in terms of Wold-type decomposition determined by T and growth of iterates of T at x. We revisit Rochberg's theorem and prove a result for isometries. When T is merely a contraction, x is a coboundary under an additional assumption. Some applications to L2-solutions of the functional equation f(x) - f(2x) = F(x), considered by Fortet and Kac, are given.