Contiguous digit sets and local roundings

The paper shows that certain interesting properties of roundings and representations hold at a fairly general level. The author restricts his attention to roundings of real numbers. He defines roundings by means of two parameters for each basic interval of the screen. These parameters determine the dividing point of the interval and the direction in which the dividing point moves under rounding. He shows that radix systems in which truncation and lexicographic ordering obey natural monotonicity conditions can be characterized as systems with contiguous digit sets. An examination is made of the interplay of roundings and representations.