Explaining why a function is well-defined How can I explain that a function is well-defined, if it's defined recursively by specifying $f(1)$, and a rule for finding $f(n)$ from $f(n-1)$?
My reasoning: If the function for $f(n)$ can be derived from $f(n-1)$, then the function must give a unique value for each input, which is part of what being well-defined is. And since I have $f(1)$, then I can prove any other function starting with that one, so it IS well-defined.

Here’s a much stronger version of the recursion theorem then the one presented in Arturo Magidin’s answer.

Let $X$ be a set,  let $X^{<\omega}$ denote the set of all finite sequences of elements of $X$ , let $[n]$ denote $\{0,1,2,…,n-1\}$ for $n\in \omega$. Suppose $g:X^{<\omega}\to X$ then there exists a unique function $f:\omega \to X$ such that $f(n)=g(f\restriction [n])$ for all $n \in 
\omega $.

One  can prove existence and uniqueness quite easily , by building approximations to $f$, and then “pasting them together”.This version is much stronger, because it “allows access” to $(f(0),f(1),f(2),…,f(n-1))$. This also  lets us handle things like the Fibonacci sequence and much more.
