A lookup table is simply one far end of a spectrum of possible ways to implement some map from inputs to outputs.

Yes. In my view, this continuity is best observed through graph-theoretic properties of various finite state machines that implement the same mapping of inputs to outputs (since every computation that occurs in reality must be in the form of a finite state machine). From this perspective, the lookup table is a very sparse graph with very many nodes, but there's nothing special about it otherwise.