The limiting values of smoothly-convex monotonic functions and properties of infinity

An introduction contains the short methodological agreement on symbols and on terms, on concepts and on mathematical texts. Then we have find the border between a finite and an infinite subset of the linearly ordered set. By definition each subset of the final set (except trivial) has two boundary points. The set is called as an infinite set if there [{exits}] its subset, which has less two boundary points. Further we have established some facts of the theory of smoothly convex monotonous functions. In particular, smoothly convex monotonous functions have the nonnegative first derivative and they are not limited. In item 3 we formulate the alternative extension of the real numbers set. In the conclusion we note that the class of bijections on the real numbers set consists of sectionally linear functions with single slope.