Theorem: A computable universal function for the class of functions of $n$ variables exists that are defined on a finite subset of $\mathbb N^n$.
Attempt at proof: Each such function is completely described by a finite table of pairs of its arguments $x_i$ and values $f(x_i)=f_i$. Assume $P, P^n,k$ are primitive recursive encodings of the pairs, the $n$-tupples, and the lists respectively. Then encode each table $f$ by $c(f)=k(<P(f_1,P^n(x_1)),\dots,P(f_m,P^n(x_n))>$). This gives us a sequence $\{\phi_i\}^\infty$ of all such functions. Define $\Phi$ by $\Phi(i,x)=\phi_i(x)$. This is a universal function.
Problem: $\Phi$ is apparently computable, for each function defined on a finite subset is the restriction of a polynomial. I am trying to decode the Lagrange interpolation multinomial for a given index of the sequence, but I'm having major difficulties re-constructing it for any given index.
Can I get some help with that, or with reducing the problem to uni-variate Lagrange polynomials?
You do not need interpolating polynomials, unless you really need to stick to the two cases when your function is defined or undefined. If you are willing to brake further, you may just specify each value in a separate case. Then $\Phi$ is obviously computable, for the constants are computable and the cases are computable.