Functions on Okounkov bodies coming from geometric valuations (with an appendix by S\'ebastien Boucksom)

We study topological properties of functions on Okounkov bodies as introduced by Boucksom-Chen and Witt-Nystr"om. We note that they are continuous over the whole Okounkov body whenever the body is polyhedral, on the other hand, we exhibit an example that shows that continuity along the boundary does not hold in general.