A Missing Proofs A.1 Proof of Theorem 1, Intra Order-preserving Functions Theorem 1. A continuous function f: R

By Lemma 1, we know ˆ f is continuous and therefore w is also continuous. U w ( y), where U, w, and y are in Theorem 1. To prove Theorem 2, we first study the properties of order invariant functions in Appendix A.2.1. Theorem 1 to prove Theorem 2 in Appendix A.2.2. In fact, every order invariant function is equality-preserving.