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

the problem is to decide if program $P_x$ has an infinite range, meaning that it outputs an

infinite number of different values. Let $d_{IR}(x)$ be the decision function for Infinite Range

and consider the following function $g(x)$ which attempts to diagonalize against all computable

functions in an attempt to prove that $d_{IR}(x)$ is not total computable.

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

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.