Generating Sequences of Functions

In 1934 Sierpiński [5] proved that every function from a countable family X of continuous functions f : [0, 1] → [0, 1] can be obtained as a composition of four such functions. These four functions are said to generate the aforementioned family. The sequence X is not necessarily closed under composition, so it may be more precise to say that these four functions generate a semigroup containing X . The difference being minor, we allow ourselves this abuse of notation. The purpose of this paper is to find the least number of functions from a family F that generate any given sequence of functions from F . The families considered are continuous, Baire-n, Lesbegue or Borel measurable, increasing or differentiable functions from the closed unit interval [0, 1] to itself and increasing functions from the natural numbers N to N. A not entirely up-to-date survey of the rich history of the type of result considered in this paper can be found in [4]. Other classical theorems can be found in [1], [2], [3] and [6]. In Theorem 2.1 we prove, using elementary arguments, that it is possible to generate any countable family of continuous functions using just two such functions. Finding two continuous functions which cannot be generated by one continuous function is not difficult. The argument used for continuous functions is modified to show the analogues for Baire-n functions and Lebesgue measurable functions in Theorems 2.4 and 2.5, respectively. With three examples of families of functions with this property, it is reasonable to ask for an example where the number of functions required is not 2. Theorem 3.1 demonstrates that every function from a countable family of increasing functions on the closed unit interval can be generated by three, but not necessarily two, such functions. On the other hand, there exists a countable family of increasing functions on the natural numbers that is not generated by any finite number of such functions: see Theorem 4.1. Finally, in Theorem 4.2 a countable sequence of infinitely many times differentiable functions is given that is not generated by any finite number of differentiable functions. The convention of writing mappings on the left, with composition from right to left, is followed in this paper. Simple juxtaposition fg is used to denote the composition f ◦ g of functions f and g.