Let S denote the set of all total, unary, computable functions from N to N. Since S is denumerable, we can write S = {f0, f1, f2, … }. Define g : N -> N by g(x) = fx(x) + 1. Then g is different from every fx, so g is not in S.
On the other hand, we see as follows that g is in S. Since fx is total for every x, fx(x) + 1 is defined for every x, so g is total. It remains to show g is computable. Given any x, first compute fx(x); we can do this since fx is computable and total. Then add 1 and output the result as the value of g(x). So, by Church’s Thesis, g is computable.