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 (A1). For an indirect proof of (A2), assume that
which contradicts the continuity of
(Ü) It follows from (A1)
implying that
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 (B1). Suppose the contrary to (ii), there is
is nonexistence since
(Ü) The condition (B1) implies
and
Thus, (6) and (7) lead to
which implies that the jumping discontinuous point
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
(C1)
(C2)
Proof. (Þ) We first show that the condition (C1) holds. Suppose the contrary, for any
corresponding point
contradiction. This gives a proof to (C1). To prove (C2), by reduction to absurdity, we assume that
is nothingness, which contradicts the continuity of
(Ü) From the assumption (C1) we see that
implying the oscillating discontinuous point
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
is infinite, which contradicts the continuity of
(Ü) From the assumption (D1) and the fact
implying the infinite discontinuous point
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).