Consider the decision problem Empty which takes as input a G$$del number $x,$ and the problem

is to decide if program $P_x$ has an empty range, meaning that $P_x$ does not halt on any input.

Let $d_{\text{E}\text{m}\text{p}\text{t}\text{y}}(x)$ be the decision function for Empty and consider the following function $g(x)$

which attempts to diagonalize against all computable functions in an attempt to prove that

$d_{\text{E}\text{m}\text{p}\text{t}\text{y}}(x)$ is not total computable.

$g(x)=\{\begin{array}{ccc}1& if& d_{\text{E}\text{m}\text{p}\text{t}\text{y}}(x)=1\\ \text{u}\text{a}\text{r}\text{r}& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

Explain why it is possible that there is an index i for which $g(x)={}_i(x).$ In other words, it is

possible that $g$ may fail to diagonalize against all computable functions.