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This paper investigates four classes of functions with a single discontinuous point. We give the sufficient and necessary conditions under which the second order iterates are continuous functions. Furthermore, the sufficient conditions for the continuity of the even order iterates with finitely many discontinuous points are obtained.

For a nonempty set X and

of functions [

The purpose of this paper is to study the discontinuous functions defined on open interval. For four classes of discontinuous functions with unique discontinuous point, we obtain the sufficient and necessary conditions for functions being continuous ones under second iterate, which are easily verified respectively. As corollaries, the sufficient conditions for the continuity of the even order iterates with finitely many discontinuous points are obtained. Our results are illustrated by examples in Section 3 .

In this section the main results for the continuity of

Theorem 1. Suppose that

Then

Proof. (Þ) Assume that

On the other hand, using the definition of

Thus (2) and (3) lead to (A_{1}). For an indirect proof of (A_{2}), assume that

which contradicts the continuity of _{2}).

(Ü) It follows from (A_{1})

implying that _{2}), i.e.,

Corollary 1. Suppose that

()

()

are fulfilled for all

Proof. By using the sufficiency of Theorem 1, the assumption (

Thus

Theorem 2. Suppose that

Then

Proof. (Þ) In view of the definitions of

and

Clearly, (4) and (5) yield (B_{1}). Suppose the contrary to (ii), there is

is nonexistence since _{2}).

(Ü) The condition (B_{1}) implies

and

Thus, (6) and (7) lead to

which implies that the jumping discontinuous point _{2}) in Theorem 1, we can prove that all points

Corollary 2. Suppose that

()

()

are fulfilled for all

Proof. The discussion is similar as that of Corollary 1. By using the sufficiency of Theorem 2, the assumption (

Theorem 3. Suppose that

(C_{1})

(C_{2})

Proof. (Þ) We first show that the condition (C_{1}) holds. Suppose the contrary, for any

corresponding point

contradiction. This gives a proof to (C_{1}). To prove (C_{2}), by reduction to absurdity, we assume that

is nothingness, which contradicts the continuity of_{2}) is proved.

(Ü) From the assumption (C_{1}) we see that

implying the oscillating discontinuous point _{2}) in Theorem 1 and prove that all points

Corollary 3. Suppose that

(

()

are fulfilled for all

Proof. The discussion is similar as that of Corollary 1. By using the sufficiency of Theorem 3, the second iterate

Theorem 4. Suppose that

Proof. (Þ) Note that

which shows the limit _{1}). To prove (D_{2}), suppose the contrary, there exists a point

is infinite, which contradicts the continuity of_{2}) is completed.

(Ü) From the assumption (D_{1}) and the fact

implying the infinite discontinuous point _{2}) holds, then all real numbers

Corollary 4. Suppose that

()

()

Proceeding similarly as Theorem 4 one can show this corollary.

Corollary 5. Suppose that

()

()

are fulfilled for all

Proof. We obtain the result by using the similar argument as Corollary 1. In view of the sufficiency of Theorem 4, the second iterate

all points

In this section we demonstrate our theorems with examples.

Example 1. Consider the mapping

Clearly,

Moreover, the set

Example 2. Consider the mapping

Clearly,

and

Example 3. Consider the mapping

Clearly,

Example 4. Consider the mapping

Clearly,

and

We thank the Editor and the referee for their comments. Project supported by Shandong Provincial Natural Science Foundation of China (ZR2014AL003), Scientific Research Fund of Sichuan Provincial Education Departments (12ZA086), Scientific Research Fund of Shandong Provincial Education Department (J12L59) and Doctoral Fund of Binzhou University (2013Y04).