Background:
In math class, the teacher was explaining recursion with examples like the Fibonacci sequence and the gamma function and proved how both examples can be written recursively. He taught us that a recursive formula is isomorphic to a map from $T \rightarrow U$, and a function is isomorphic to a map from $U \rightarrow T$ and the representation of such formulas can be written in lambda calculus. We only have studied functions with subsets of N as the domain
Question:
Can you turn all functions into a recursive formula?
Instinctively I thought not, but I was wondering if someone could provide any insight.
No; consider all functions from $\mathbb{N} $ to $ \{0, 1\} $. By Cantor's famous diagonal argument, there is an uncountable number of such functions