We extend some of the classical connections between automata and logic due to Buchi and McNaughton and Papert, to languages of finitely varying functions or "signals". In particular we introduce a natural class of automata for generating finitely varying functions called ST-NFA's, and show that it coincides in terms of language-definability with a natural monadic second-order logic interpreted over finitely varying functions. We also identify a "counter-free" subclass of ST-NFA's which characterize the first-order definable languages of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic (MTL).