Continuity Function on Partial Metric Space

Corresponding Author: Fitri Aryani Department of Mathematics, Faculty of Science and Technology, UIN Suska, Pekanbaru, Riau, Indonesia; Email: a.fudholi@gmail.com khodijah_fitri@uin-suska.ac.id Abstract: Ordered pairs form of a metric space (S,d), where d is the metric on a nonempty set S. Concept of partial metric space is a minimal generalization of a metric space where each x∈S,d(x,x) does not need to be zero, in other terms is known as non-self-distance. Axiom obtained from the generalization is following properties p(x,x)≤p(x,y) for every x,y∈S. The results of this paper are few studies in the form of definitions and theorems concerning continuity function on partial metric space.


Introduction
Lately, there has been a lot of mathematicians interested in developing the study of metric spaces that appear various generalization of metric spaces. One generalization of a metric space is a partial metric space developed by Matthews in 1992. Several studies on partial metric space has ever done is Romaguera and Schellekens (2005) researching on the basic concept of a partial metric space in the Quantitative Domain Theory in journals titled "Partial monoids and Semivaluation Metric Spaces". Subsequently, Wahyuni (2012) in the journal entitled "Topology of Partial Metric" examines the topology built by base ball open partial metric.
Subsequently, in 2014 Devi Arintika have discussed about Banach fixed point theorem applicable on a partial metric spaces in his journal, entitled "Generalitation Banach Fix Point Theorem on Partial Metric Space". Then, 2015 Ge Xun and Lin Shou examines the existence and uniqueness theorems for completion of a partial metric space in the journal entitled "Completion of Partial Metric Spaces".
Based on these reviews, the authors are interested to observe about one generalizations of metric space that is partial metric space and concepts of continuity of partial metric space. The partial metric space will be observed in this study is:

Preliminaries
Metric and Metric Space Definition 1. (Thomson et al., 2008) Let S is nonempty set. A function: Called metric if it satisfies the following 4 properties: If d is metric on nonempty set S, then S called metric space, as described in the following definition: Definition 2. (Thomson et al., 2008) Metric space is a set of pairs (S,d), where S is nonempty set and d is a metric on S.
Definition metric and metric space more easily understood by observing the following example: x,y∈S must following four properties from Definition 1. Properties (a), (b), (c) already clearly fulfilled for (x,y) = |x-y|, while for properties (d) can be provided by using the triangleinequality in absolute value. Consider that:

Partial Metric and Partial Metric Space
Partial metric space is a generalization of metric spaces. Consider the following definitions: Definition 3. (Romaguera and Schellenkens, 2005) A partial metric on nonempty set S is a function : p S S + × → ℝ , such that for every x,y,z∈S satisfy the following axioms: According to Malhotra et al. (2014) says that a partial metric space is a pair (S,d) such that S is nonempty set and p is metric partial.
The function on the partial metric is a generalization of the axiom minimal metrics such that for every x∈S, d(x,x) does not need to be zero, in other terms is known as nonzero self-distance. Axiom obtained from the generalization is following properties p(x,x) ≤ p(y,x) (Wahyuni, 2012). Definition of partial metrics and partial metric spaces more easily understood by observing the following example: , 2 x y x y p x y − + + = , x y ∀ ∈ R . Show that (S,p) is partial metric space.

Solution
Given a nonempty set S = R with : x y x y p x y − + + = , for all , x y ∈ R will be shown that (S,d) is partial metric space, means nonempty S with function : x y x y p x y − + + = , for all , , x y z ∈ R must following four properties from Definition 3. Properties (i), (ii), (iii) and (iv) already clearly fulfilled for ( , ) 2 x y x y p x y − + + = , so (S,p) is partial metric space.
Furthermore, the definition will be given a sequence converging on a partial metric space.
Definition 4. (Arintika, 2012) A sequence {x n } in partial metric space (S,p) convergent to c∈S, for each ε >0 there N ∈ ℕ such that for all n≥N satisfy:

Continuous Function on Partial Metric Space
Before defining the concepts of continuous functions on a metric space partial, will be determined beforehand definition of the partial metric subspace, which is as follows:

Definition 5
If (S,p) is partial metric space and A⊆S, then (A,p) called partial metric subspace (S,p).
The following definitions will explain the function which is continuous of partial metric space at a point.

Definition 6
Let (S 1 ,p) and (S 2 ,p) is partial metric space, A⊆S 1 , function f:A→S 2 and c∈A.  Proof that f only have one limit point at c (uniqueness of limits).
After we get concepts ofthe limit of a functionon partial metric space, then we will observe about concepts of continuous functions on partial metric space. The following definition explain about continuous function on partial metric space at a point.

Definition 7
Let (S 1 ,p) and (S 2 ,p) arepartial metric space, A⊆S 1 , function f: A→S 2 and c∈A. f continueousat c, if for each ε >0 there exist δ >0 such that if x∈A: , To understand continuity of partial metric spaces, will be given in the following example:

Example 3
Given (S 1 ,p) and (S 2 ,p) is partial metric space and A⊆S 1 . Show that function f:A→S 2 defined on A with f(x) = x is continues.

Solution
Given f(x) = x, will be shown for any ε >0 there is δ , , . p Next, will be given a theorem about the function which is continuous in a sequence on partial metric space.

Theorem 2
Let (S 1 ,p) and (S 2 ,p) is partial metric space, A⊆S 1 , c∈A and function f: A→S 2 . The following statement is equivalent: for each n ∈ ℕ . Therefore, sequence {x n } is Bounded.

Conclusion
Based on the main results, obtained some conclusions as follows: • If (S,p) is partial metric space and A⊆S, then (A,p) is called partial metric subspace (S,p) • Some of the concept of continuity function on partial metric space is as follows: • Let (S 1 ,p) and (S 2 ,p) arepartial metric space and A⊆S 1 . If f:A→S 2 and c is limit point of A, then f only have onelimit point at c (uniqueness of limits) • Let (S 1 ,p) and (S 2 ,p) is partial metric space, A⊆S 1 , c∈A and function f:A→S 2 . The following statement is equivalent: • f continous at c • For any sequence {x n } at A convergent to c∈A, then sequence {f(x n )} convergent to f(c) • A sequence {x n } convergent of partial metric space is bounded