Real functions computable by finite automata using affine representations

This paper tries to classify the functions of type I″ →I (for some interval I ⊆ ℝ) that can be exactly realized by a finite transducer. Such a class of functions strongly depends on the choice of real number representation. This paper considers only the so-called affine representations where numbers are represented by infinite compositions of affine contracting functions on I. Affine representations include the radix (e.g. decimal, signed binary) representations. The first result is that all piecewise affine functions of n variables with rational coefficients are computable by a finite transducer which uses the signed binary representation. The second and main result is that any function computable by a finite transducer using an affine representation is affine on any open connected subset of I″ on which it is continuously differentiable. This limitation theorem shows that the set of finitely computable functions is very restricted.