Consider a finite state set $Q$, with a distinguished start state $s\in Q$, as well as two functions: a transition function $\delta:Q\times\mathbb{R}\to Q$ and a final output function $F:Q\to\mathbb{R}$. The tuple $(Q,s,\delta,F)$ defines a mapping $f:\mathbb{R}^*\to\mathbb{R}$ --- that is, from finite sequences of reals to reals --- in exactly the same way as a classic DFA over a finite alphabet $\Sigma$ defines a mapping $\Sigma^*\mapsto\{0,1\}$.
Have such DFA analogs over the real alphabet been considered before?