example of a function that could only be defined recursively is there a function that can be proved is only defineable recursively? the converse seems to be trivially false, i.e. every iteritive function is trivially defined recursively with 0 for coefficients for antecedal terms, how ever can a function such as $f(x)=x$ be defined non trvialy recursive in $\mathbb R$? $x_n = x_{n-1}+1$ defines it recursively over $\mathbb N$.
Is there a list of known ways that functions can be defined? for example consider followings:
$f(x)= \text{some algebraic expression , finite or infinite}$ : algebraicly defined
$x_{n+1}= \text{some algebraic expression involving }x_n, x_{n-1}, \cdots$ : recursively defined
$y : \text{if there are no 2 in decimal digits of x then 1 else 0} $ : defined by testing
or is there a reason not to bother with how functions defined?

In principle it is possible to "simulate" a recursive function by implementing a stack.
