Let Λ be a discrete subset of the real line ℝ. We prove that for every bounded function φ on Λ there exists an operator Lipschitz function f on ℝ such that f’ (t) = φ(t) for all t∈Λ. The same is true for the set of operator Lipschitz functions f on ℝ such that f’ coincides with the non-tangential boundary values of a bounded holomorphic function on the upper half-plane. In other words, for every bounded function φ on Λ there exists a commutator Lipschitz function f on the closed upper half-plane such that f’ (t) = φ(t) for all t∈Λ. The same is also true for some non-discrete countable sets Λ. Furthermore, we consider the case where Λ is a subset of the closed upper half-plane, Λ⊄ ℝ. Similar questions for commutator Lipschitz functions on a closed subset F of ℂ are also considered.