Approximation in Real Contra-Continuous Functions Spaces

ABSTRACT

In the real numbers with usual topology (in general) not just an approximation of these functions is not study but there is no any example for contra-continuous function because impossible finding examples for this kind in real numbers(R) since in this case some elements in a domain of function be have more than one image in its codomain as I will explain later.
As defined it previously , the function :  ⟶  is called contra-continuous function if the invers image of any an open set in  is a closed set in  (where I will used the real numbers R with usual topology (,   ) instead of the sets ,  ) and symbolled of the real contracontinuous function space by () .
As described (or define) earlier of the continuous function as the following "If :  ⟶  and let  ∘  then we say that  is continuous in So, for the contra-continuous function I will describe (or definition) it as "If :  ⟶  and let  ∘  then we say that  is contra-continuous function in Or "If :  ⟶  and let  ∘  then we say that  is contra-continuous function in

) (Note that numbering includes two parts of above results).
This definition (or description) come from definition of a closed set in (,   ) that since an open set in  will be as an open interval then its complement must be a closed set and it will be as a closed intervals or countable sets whether finite or infinite sets with condition that these numbers must be bonded.
So after defined contra-continuous function  as above then one have the right to ask especially (about studying the possibility of an approximation of this kind of functions) is this function has a norm?before answered this question there is another question for this function that is "is  bonded ?" and I will study this case with some expansion in theorem 2.3 this by used properties of convergent of sequences in domain of .Now, after proved that  bonded function ,So sequentially it has a norm and as in case of continuous functions which has two types of norms that is integral norm and supremum norm, Also for elements in a closed interval, every element can be content in a closed set(disc) which located in our interval except endpoints which only be contained in our closed interval, so every element in a closed set is located in a closed set which continent in our closed set Let's symbolize to this results by the number…………………….….. (1-3).From our information the infinite union of a closed sets not necessary a closed set but if the closed sets in (,   ) as form closed interval or countable set then the infinite union of them is a closed set as shown in results (1-2) and (1-3).We return to the approximation of the contra-continuous functions, initially I will study existences of best approximation element of these functions.It is known that the compact set give this element for any continuous function which content in it, the main question here is that" is this feature available and valid for contra-continuity functions?In this paper I study availability to get a best approximation element in the compact set.
After getting the best approximation element and an example for real contra-continuous one has the right to ask can anyone approximate the contra-continuous function by real polynomial of degree  ?I talk here of one of the well-known polynomials , such as Bernstein's operator, It is known that Bernstein's polynomials is one of the excellent polynomials that provides the best approximation elements for any continuous function within the closed interval [0,1] or its expansion, in this paper I will approximate real contra-continuity function  by Bernstein operator as in theorem 3.3.

2.Materials and Methods
For contra-continuous function, there are two questions: is there a relationship between our previous description of it and the possibility of moving the inverse image of an open set to a closed set?Here I will answer this question in the first theorem.Furthermore, does contra-continuous function bounded?Also will answered these questions in the next theorem, before that I will prove convergent of image sequence in contra-continuous function.

3.Results Discussion
After contra-continuous function was introduced and prefaced as above, here I study an approximation of this type of function.
First of all, and after I prove that contra-continuous function is bounded, and then it has a norm, after this preparing of this base I will study an approximation of this function.
I start with possibility to get best approximation element.It known that the compact set provide this element for any continuous function belong in it.So, Is this case remainder for contra-continuity functions?This is what I will answer in first part.
After studying the possibility of obtaining the element of best approximation I try to finding example for polynomial of best approximation of the innovation example for contra-continuous function.
Theorem 3.1 :[7] Let (X; d) be a metric space.Then K ⊂ X is compact if and if every sequence in K has a subsequence converging to a point in K. Theorem 3.2 : Suppose that :  ⟶  is contra-continuous function , ⊆ () ⊆  ,  is a compact set, then  is provide best approximation element for .Proof: Suppose that    such that   →  ∀ and (  ) ∈ .This mean that ∃ ≥ 0 such that |  − | ≤  Since  is contra-continuous function , then by (1-1) ∃ ≥ 0 |(  ) − ())| <  This mean that f(  ) ⟶ () Since  is compact in  then by theorem 3.1 must be () ∈  Now, if ∃ ∘  such that ∄  ⊆  and   ↛  ∘ This mean that ∃ ≥ 0 such that |  −  ∘ )| ≥  Since  is contra-continuous function , then ∃ ≥ 0 |(  ) − ( ∘ ))| >  But this contradict with  is bonded So, (  ) ⟶ () ∈ , ∀As  → ∞ And then  has a best approximation element in ■ Now for an example about real contra-continuous function, it is clear that this example impossible to get it in real numbers because there will be more than one elements has two images (or more than) in codomain of this function as mentioned earlier.