Some characterizations of functions computable in on-line arithmetic

After a short introduction to on-line computing, we prove that the functions computable in on-line by a finite automaton are piecewise affine functions whose coefficients are rational numbers (i.e., the functions f(x)=ax+b, or f(x,y)=ax+by+c where a, b, and c are rational). A consequence of this study is that multiplication, division and elementary functions of operands of arbitrarily long length cannot be performed using bounded-size operators