The Root of Revenue Continuity

In the setup of selling one or more goods, various papers have shown, in various forms and for various purposes, that a small change in the distribution of a buyer's valuations may cause only a small change in the possible revenue that can be extracted. We prove a simple, clean, convenient, and general statement to this effect: let $X$ and $Y$ be random valuations on $k$ additive goods, and let $W(X,Y)$ be the Wasserstein (or "earth mover's") distance between them; then $$\left\vert \sqrt{Rev(X)}-\sqrt{Rev(Y)}\right\vert \le \sqrt{W(X,Y)}.$$ This further implies that a simple explicit modification of any optimal mechanism for $X$, namely, "uniform discounting," is guaranteed to be almost optimal for any $Y$ that is close to $X$ in the Wasserstein distance.