Preimage cardinalities of continuous functions
Introduction
It is a well-known fact that there exists no continuous function such that every point has either two or zero preimages. A somewhat less well-known fact is that there exist continuous functions where the preimage has an even cardinality for all , in fact, one can construct such a function where the set of cardinalities of preimages is .
In this paper, we are interested in the question of what other sets may occur as the set of preimage cardinalities for a continuous function. Given a continuous function we let Our question is what subsets of 1 may occur as where is a continuous function. Notice that by boundedness of a continuous function in the compact interval, it is necessary that . We also study the case when f belongs to a sub-class of the class of continuous functions.
One can also ask what subsets S of occur as but in this question if we let (otherwise, we fall back to the previous question) then it becomes easy: for any non-empty subset where , the set does occur as for some continuous ; we leave this as an exercise for the reader (notice that since , one can allow f to be constant on certain sub-intervals of ).
By re-scaling the domain, we may just consider functions over the interval in this question (instead of an arbitrary interval). For an arbitrary interval , will denote the space of all continuous functions . In this space, we define a subset In the case of the interval , we will just write instead of . For functions , we can also write If , then we also say that f is an S-function. Notice that a function may have infinitely many local maximums and local minimums, moreover, the end points 0 and 1 are either a local maximum or a local minimum.
Given a subset with , we let Here, we use a convention that the supremum of an unbounded subset of natural numbers equals ∞. Our main theorem is the following
Theorem 1 Let such that and . Then there exists with if and only if .
With slight care, one can indeed make all these functions from the class . An important tool in this construction is the use of functions which “wiggles” infinitely many times. Hence this construction would not work for real analytic functions (by our definition, a function is real analytic if it can be extended holomorphically (see [1]) to some domain of which includes the unit real segment ; in the sequel, instead of “real analytic” we will simply use the term “analytic”). Motivated by this fact, we also study the same question for all analytic functions, in particular, for polynomials. In fact, by far, not every subset of type for a continuous function occurs as where p is a polynomial. First of all, for a polynomial , we necessarily have is finite (same is true for analytic functions ). But even among finite subsets, for example, the set already does not occur as . By the following theorem, we achieve a complete characterization of the sets as well where is an analytic function. To state this theorem, we need to introduce some notations. Let
Theorem 2 Let such that . Then there exists an analytic function with if and only if there exists a finite sequence of natural numbers for some such that the following conditions hold: for all , and for all there exists such that , and , for all , either or , if any triple belongs to B, then no triple belongs to C.
Moreover, if conditions i)-v) hold, then there exists a polynomial with .
Section snippets
Proof of Theorem 1
In this section we will assume that S is a subset of such that and . The symbols will always denote a set for some or one of the sets . Such sets will be called index sets. We will first consider the special case of Theorem 1 when . In this special case Theorem 1 states the following
Proposition 1 Let such that . Then there exists with .
To prove Proposition 1, we introduce some terminology.
Definition 1 Let . A function is called an n-wave
Proof of Theorem 2
Let us first recall a well-known fact that if an analytic function has infinitely many zeros on a compact interval then it is identically zero. Thus a non-constant analytic function has finitely many zeros and finitely many critical points. It also belongs to . Then there exists a finite sequence such that any local maximum or local minimum point of p belongs to , and each of the subsets contains at least one local maximum or local minimum. Let
Acknowledgement
I thank Azer Akhmedov for his encouragements and for so many helpful comments and suggestions on the original draft. I also thank the anonymous referee for pointing out several inaccuracies in the original version and for making a number of very useful suggestions.
References (2)
Invitation to Complex Analysis
(2010)- et al.
Lagrange's formula of interpolation