 On the assumption that \(h(x,y)\) is computable, \(g(x)\) is partial computable since, e.g., it may be computed by a program which on input \(x\) computes \(h(x,x)\) and returns 0 just in case \(h(x,x) = 0\) and otherwise goes into an infinite loop. It hence follows that \(g(x) \simeq \phi_j(x)\) for some \(j \in \mathbb{N}\). But now observe that one of the following two alternatives must hold: i) \(g(j) \darrow\); or ii) \(g(j)\uarrow\). We may thus reason by cases as follows: