# Continuous Regular Functions

## TL;DR

This paper characterizes continuous r-regular functions as locally affine except on a null set and shows differentiable r-regular functions are globally affine, linking automata theory with geometric properties.

## Contribution

It proves that continuous r-regular functions are locally affine outside a null set and that differentiable ones are affine, connecting automata with geometric analysis.

## Key findings

- Continuous r-regular functions are locally affine outside a null set.
- Differentiable r-regular functions are globally affine.
- Deciding differentiability of r-regular functions is in PSPACE.

## Abstract

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.