[proofplan] We prove continuity by checking preimages of open subsets of the discrete two-point space $\{0,1\}$. The preimage of $\{1\}$ is the basic clopen set $[\varphi]$, and the preimage of $\{0\}$ is the basic clopen set $[\neg\varphi]$. Since every open subset of $\{0,1\}$ is a union of the sets $\varnothing$, $\{0\}$, $\{1\}$, and $\{0,1\}$, these computations prove continuity. [/proofplan] [step:Identify the two nontrivial point preimages] Fix an $A$-formula $\varphi(x)$. Define the formula evaluation map \begin{align*} f_\varphi: S_x(A) &\to \{0,1\} \\ p &\mapsto \begin{cases} 1, & \varphi \in p, \\ 0, & \varphi \notin p. \end{cases} \end{align*} By the definition of $f_\varphi$, \begin{align*} f_\varphi^{-1}(\{1\}) &= \{p \in S_x(A) : f_\varphi(p) = 1\} \\ &= \{p \in S_x(A) : \varphi \in p\} \\ &= [\varphi]. \end{align*} Because each $p \in S_x(A)$ is a complete type, exactly one of $\varphi$ and $\neg\varphi$ belongs to $p$. Hence \begin{align*} f_\varphi^{-1}(\{0\}) &= \{p \in S_x(A) : f_\varphi(p) = 0\} \\ &= \{p \in S_x(A) : \varphi \notin p\} \\ &= \{p \in S_x(A) : \neg\varphi \in p\} \\ &= [\neg\varphi]. \end{align*} [guided] Fix an $A$-formula $\varphi(x)$. The map whose continuity we want to prove is \begin{align*} f_\varphi: S_x(A) &\to \{0,1\} \\ p &\mapsto \begin{cases} 1, & \varphi \in p, \\ 0, & \varphi \notin p. \end{cases} \end{align*} The natural way to prove continuity into a discrete two-point space is to compute the preimages of the two singleton sets. First consider the point $1 \in \{0,1\}$. By the definition of inverse image and then by the definition of $f_\varphi$, \begin{align*} f_\varphi^{-1}(\{1\}) &= \{p \in S_x(A) : f_\varphi(p) \in \{1\}\} \\ &= \{p \in S_x(A) : f_\varphi(p) = 1\} \\ &= \{p \in S_x(A) : \varphi \in p\}. \end{align*} The last set is precisely the basic Stone set determined by $\varphi$, namely \begin{align*} [\varphi] := \{p \in S_x(A) : \varphi \in p\}. \end{align*} Therefore \begin{align*} f_\varphi^{-1}(\{1\}) = [\varphi]. \end{align*} Now consider the point $0 \in \{0,1\}$. Again, by the definition of inverse image and by the definition of $f_\varphi$, \begin{align*} f_\varphi^{-1}(\{0\}) &= \{p \in S_x(A) : f_\varphi(p) = 0\} \\ &= \{p \in S_x(A) : \varphi \notin p\}. \end{align*} To rewrite this set in Stone-basic form, we use completeness of types. Since $p$ is a complete type over $A$, for the formula $\varphi(x)$ it contains exactly one of $\varphi$ and $\neg\varphi$. Thus, for every $p \in S_x(A)$, \begin{align*} \varphi \notin p \quad \iff \quad \neg\varphi \in p. \end{align*} Substituting this equivalence into the description of the preimage gives \begin{align*} f_\varphi^{-1}(\{0\}) &= \{p \in S_x(A) : \neg\varphi \in p\} \\ &= [\neg\varphi]. \end{align*} [/guided] [/step] [step:Use the Stone topology and discreteness of $\{0,1\}$ to prove continuity] By definition of the Stone topology on $S_x(A)$, the sets $[\varphi]$ and $[\neg\varphi]$ are open. Therefore $f_\varphi^{-1}(\{1\})$ and $f_\varphi^{-1}(\{0\})$ are open. Since $\{0,1\}$ has the discrete topology, its open subsets are $\varnothing$, $\{0\}$, $\{1\}$, and $\{0,1\}$. The remaining two preimages are \begin{align*} f_\varphi^{-1}(\varnothing) &= \varnothing, \\ f_\varphi^{-1}(\{0,1\}) &= S_x(A), \end{align*} which are open in $S_x(A)$. Thus the preimage under $f_\varphi$ of every open subset of $\{0,1\}$ is open in $S_x(A)$. Hence $f_\varphi$ is continuous. [/step]