 Intuitively, if \(f\) represents \(\Phi\) then \(f(\alpha(\overline{n}))=0\) means that \(\alpha(\overline{n})\) is not long enough to compute \(\Phi(\alpha)\), and \(f(\alpha(\overline{n}))=m+1\) means that \(\alpha(\overline{n})\) is long enough to compute \(\Phi(\alpha)\) and that the value of \(\Phi(\alpha)\) is \(m\). If \(\mathcal{K}\) denotes the class of neighborhood functions, then the continuity axiom \({\bf C\mbox{-}N}\) can be rephrased as   \[  \forall \alpha\exists n A(\alpha,n) \rightarrow   \exists f \in \mathcal{K}\, \forall m(f(m) \gt 0 \rightarrow   \forall \beta \in m A(\beta,f(m-1))),  \]   