# computability – Given a total recursive function, can you still calculate its fixed point?

We know from Kleene's recursion theorem that if $$f$$ is recursive total, there must be an integer $$n$$ For who $$varphi_n = varphi_ {f (n)}$$. My question is: for every $$f$$ recursive total, is there a computable (total) procedure that computes such a point?

For example, given $$f$$, for each $$x$$ we have that $$varphi_ {g (x)} = varphi_ {f (g (x))}$$ with $$g$$ recursive total.