# Why does the unbounded $\mu$ operator preserve effective computability?

Let $$f$$ be a partial function from $$\mathbb{N}^{p+1}$$ to $$\mathbb{N}$$. The partial function $$(x_1,…,x_p)\mapsto \mu y[f(x_1,…,x_p,y)=0]$$ is defined in the following way: If there exists at least one integer $$z$$ such that $$f(x_1,\dots, x_p,z)=$$ and if for every $$z', $$f(x_1,\dots, x_p,z’)$$ is defined, then $$\mu y[f(x_1,\dots, x_p,y)=0]$$ is equal to the least such $$z$$. In the opposite case, $$\mu y[f(x_1,\dots, x_p,y)=0]$$ is not defined.

I don’t understand why this unbounded $$\mu$$ operator preserves effective computability, in my textbook and in a note that I found online, this is mentioned as if it is a trivial fact.

I appreciate any help!