 Consider the domain \({\{ 0,1 \}}^\infty\) whose elements are finite and infinite sequences of bits \(0,1\), where \(u \sqsubseteq v\) if either \(u\) is infinite and \(u = v\), or \(u\) is finite and \(u\) is a prefix of \(v\). What properties should be true of a computable function taking as arguments an infinite sequence of bits \({\langle b_1,b_2,b_3,\ldots \rangle}\)? Take as an example the function \(\textit{search}:{\{ 0,1 \}}^\infty \to {\mathbb{B}}_\bot\) whose value is \({\textit{tt}}\) if, for \(u \in {\{ 0,1 \}}^\infty\), \(1\) occurs in \(u\) at least once, otherwise \(\bot\). Think of the sequence \({\langle b_1,b_2,b_3,\ldots \rangle}\) as given one element at a time: the initial segments obtained in this process are an increasing chain of finite elements of \({\{ 0,1 \}}^\infty\),  \[{\langle  \rangle} \sqsubseteq {\langle b_1 \rangle} \sqsubseteq {\langle b_1,b_2 \rangle} \sqsubseteq {\langle b_1,b_2,b_3 \rangle} \sqsubseteq \ldots\]   having \({\langle b_1,b_2,b_3,\ldots \rangle}\) as a limit (i.e., least upper bound). By monotonicity we have a corresponding increasing chain of values   \[\textit{search}({\langle  \rangle}) \sqsubseteq \textit{search}({\langle b_1 \rangle}) \sqsubseteq \textit{search}({\langle b_1,b_2 \rangle}) \sqsubseteq \textit{search}({\langle b_1,b_2,b_3 \rangle}) \sqsubseteq \ldots\]   If \(\textit{search}({\langle b_1,b_2,b_3,\ldots \rangle}) = {\textit{tt}}\), then there must be a finite initial segment \({\langle b_1,b_2,\ldots,b_n \rangle}\) for which \(\textit{search}({\langle b_1,b_2,\ldots,b_n \rangle}) = {\textit{tt}}\), and this will be the cumulative value of the function for the infinite sequence \({\langle b_1,b_2,b_3,\ldots \rangle}\). In general, a computable function \(f : D \to E\) should (be monotonic and) have the property that a finite amount of information on the output \(f(x)\) must be already obtainable by giving a finite amount of information on the input \(x\). This is equivalent to the notion of continuity originally introduced by Scott in his theory of computable functions over domains: