Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f:[0,1] to [0,1]$ is $r$-regular if there is a B{u}chi automaton that accepts precisely the set of base $r in mathbb{N}$ 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 $operatorname{PSPACE}$. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.
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