Principles of rational analysis - continuity of functions

A mathematical apparatus is proposed for constructing analysis on a set of rational numbers. Soft boundaries of sets, soft limit, soft continuity of rational functions are introduced. Extreme proximity maps for the soft continuity of the function are found. For soft continuity, analogues of the properties of classical continuous functions are proved. The concept of approximation of a real function using a rational function is introduced. It is proved that both the approximated and the approximating functions must have soft continuity properties.