zlacker

>There's a rich family of universal approximation theorems

Wow, look-up tables can get increasingly good at approximating a function!

>>sudden+(OP)
A function is by definition a lookup table.

The lookup table is just (x, f(x)).

So, yes, trivially if you could construct the lookup table for f then you'd approximate f. But to construct it you have to know f. And to approximate it you need to know f at a dense set of points.