The Riemannian Hebbarkeitss\"atze for pseudorigid spaces

We prove Riemann's theorems on extensions of functions over certain mixed characteristic analytic adic spaces, first introduced by Johansson and Newton. We use these results to reprove a theorem of de Jong identifying global sections of an $\mathcal{O}_K$-flat normal formal scheme, locally formally of finite type over $\mathcal{O}_K$, with locally powerbounded sections over the generic fibre.