Computable Classifications of Continuous, Transducer, and Regular Functions

We develop a systematic algorithmic framework that unites global and local classification problems for functional separable spaces apply it to attack concerning the Banach space C[0,1] of real-valued continuous functions on unit interval. prove problem (binary) regular among almost everywhere linear, pointwise linear-time Lipshitz is $\Sigma^0_2$-complete. show function $f\colon [0,1] \rightarrow \mathbb{R}$ transducer if only regular; interestingly, this peculiar nontrivial fact was overlooked by experts in automata theory. As one many consequences, our $\Sigma^0_2$-completeness result covers class as well. Finally, we $C[0,1]$ admits an arithmetical spaces. Our proofs combine methods abstract computability theory, analysis.