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'<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!