Simplest functions associated with operators of the left-side shift of elements of the chain image of numbers

Abstract

Properties of functions of type $\tau_????(????) = \[????(????_1(????), ????_2(????), ..., ????_????(????)), ????_{????+1}(????), ????_{????+2}(???? ), ...]$, where `$\[????_1, ????_2, ..., ????_????, ...\]$ is regular continued fraction expansion of $???? \in (0; 1]$ and $????(????_1, ????_2, ..., ????_????)$ is a natural function of natural arguments $????_1, ????_2, ..., ????_????$ are studied. Their relation with left-shift $????(????)$ and right-shift $\delta_????(????)$ continued fractions elements operators are examined. It is shown that this function are correctly defined, piecewise continuous and piecewise monotonic. Differential and integral properties of these function are also studied. Continuous transformation of a segment $[0;1]$ is constructed.