Urysohn function from closure-nested open sets ($f(x)=\inf \{t:x\in U_t\}$)

Let open sets $U_t$ be indexed by a dense subset of $[0,1]$, with $\overline{U_s}\subseteq U_t$ for $s<t$. Taking the infimum of membership indices, with value one outside their union, gives a continuous function. The strict sublevel sets are unions of $U_t$; strict superlevel sets are unions of complements of $\overline{U_t}$. These identities prove continuity.