An analytic function on open set $U\subseteq(\spec A_{Z})^{an}$ is a map $h:U\to\coprod_{z\in U}\hat{\kappa}(z)$ which is a local uniform limit of rational functions: every $z\in U$ has an open neighbourhood $U^{\prime}\subseteq U$ such that for every $\epsilon>0$, there exists $f_{U^{\prime}},g_{U^{\prime}}\in A_{Z}$ with $|h(z)-\frac{f_{U^{\prime}}(z)}{g_{U^{\prime}}(z)}|<\epsilon$ and $g(z)\neq 0$ for all $z\in U^{\prime}$. Denote by $\mathcal{R}^{\mathrm{an}}(U)$ the $k$-algebra of all analytic functions on $U$.