Continuous regular functions

Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function f: [0, 1] → [0, 1] is r-regular if there is a Büchi automaton that accepts precisely the set of base r ∈ N representations of elements of the graph of f. We show that a continuous r-regular function f is locally affine away from a nowhere dense, Lebesgue null, subset of [0, 1]. As a corollary we establish that every differentiable r-regular function is affine. It follows that checking whether an r-regular function is differentiable is in PSPACE. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.