By Proposition 2.9 (viii) it is enough to prove the continuity of ${P}_{\theta}(u)$. By the semistable reduction theorem [BL85, §7], there is a finite field extension $K^{\prime}/K$ such that $X^{\prime}\coloneqq X\otimes_{K}K^{\prime}$ has a strictly semistable model $\mathscr{X}^{\prime}$ with $\theta^{\prime}\coloneqq q^{*}\theta$ determined on $\mathscr{X}^{\prime}$, where $q\colon X^{\prime}\to X$ is the canonical map. It follows from Proposition 3.10 that ${P}_{\theta^{\prime}}(u\circ q)$ is continuous. We know from Lemma 2.11 that

$\displaystyle{P}_{\theta^{\prime}}(u\circ q)=q^{*}({P}_{\theta}(u)).$

By [Ber90, Prop. 1.3.5], the topological space of ${X^{{\mathrm{an}}}}$ is the quotient of $(X^{\prime})^{\rm an}$ by the automorphism group of $K^{\prime}/K$. We conclude that ${P}_{\theta}(u)$ is continuous. ∎