A Missing Proofs is intra order-preserving, if and only if f(x) = S(x) Uw(x) with U being an upper-triangular matrix of ones and w: R

Proof of Theorem 1. () For a continuous intra order-preserving function f(x), let w(x) = U First we show w is continuous. Because f is intra order-preserving, it holds that S(x) = S(f(x)). Let ˆf(x):= S(f(x))f(x) be the sorted version of f(x). By Lemma 1, we know ˆf is continuous and therefore w is also continuous. Next, we show that w satisfies the properties listed in Theorem 1. These two arguments prove the necessary condition.