EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5909 ISSN 1307-5543 – ejpam.com Published by New York Business Global Stieltjes Limits in Continuous Function Spaces Moch. Alifuddin1, Mahmud Yunus1,∗, Mohamad Ilham Dwi Firmansyah1, Indra Cahya Ramdani1 1 Department of Mathematics,Faculty of Sciences and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia Abstract. Sir Isaac Newton and Gottfried Wilhelm Leibniz first discovered the concept of limits in calculus in the late 17th century. Both developed calculus with different but complementary approaches and notations. The Stieltjes limit is a generalization of the standard limit by replac- ing x approaching x0 with f(x) approaching f(x0). In this paper, we define the limit and the Stieltjes limit on function-valued continuous operators, specifically, on bounded operators that take values in continuous functions. To support this definition, we first introduce the concepts of neighborhoods, continuous function operators, increasing operators, strictly increasing operators, decreasing operators, strictly decreasing operators, and their respective limits in function-valued operators. The results reveal the necessary conditions and properties of limits and Stieltjes limits on function-valued operators, which share similarities with limits and Stieltjes limits in real-valued functions. These findings broaden our understanding of limit concepts in a more general context and provide new insights into operator analysis. 2020 Mathematics Subject Classifications: 26A42, 47G10, 47B38, 46E30, 28A33 Key Words and Phrases: Stieltjes limit, Function-Valued Continuous Operator, Stieltjes limit on Function-Valued Continuous Operators, Functional Analysis 1. Introduction Mathematics continuously evolves, with integral theory being one of its fundamental branches, which undergoes ongoing research and development. Integral theory plays a cru- cial role in various mathematical applications, particularly in solving engineering problems [1]. For instance, in financial engineering, integrals are utilized in investment calculations [2]; in mechanical engineering, integrals are applied in fluid mechanics [3]; and in electrical engineering, integral analysis helps determine the current, voltage, and power in electrical circuits [4]. Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716) independently explored the fundamental concept of integrals. Their work laid the foundation for dif- ferentiation and integration, leading to the continuous advancement of integral theory, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5909 Email addresses: yunusm@matematika.its.ac.id (M. Yunus) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 2 of 17 including its modern applications [5]. One significant development in integral theory is its extension to function spaces, particularly for continuous functions [6]. In the 19th century, German mathematician Georg Friedrich Bernhard Riemann (1826 – 1866) formally defined the concept of definite integrals, even though Newton and Leibniz had already established the fundamental theorem of calculus. Due to his contributions to integral theory, the definite integral is often referred to as the Riemann integral [7]. This definition serves as the foundation for solving various mathematical problems in elementary calculus [8]. Further advancements in integral theory led to the introduction of a generalized form of the Riemann integral, known as the Riemann-Stieltjes integral, which addresses func- tions that are not Riemann-integrable [9]. This integral was first introduced by Thomas Joannes Stieltjes in 1894 [10]. The Riemann-Stieltjes integral extends the Riemann inte- gral by replacing the variable of integration x with a function r(x), thereby generalizing the concept to accommodate a broader class of functions [10]. Several studies have investigated the Riemann-Stieltjes integral with real-number bound- aries, including works by Gregory Convertito and David Cruz-Uribe [11], as well as Joong Kwoen Lee and Han Ju Lee [12]. However, this study defines the Riemann-Stieltjes inte- gral with integration limits defined by continuous functions on [a, b]. Using the ordering relation f ⪯ g, which implies f(x) ≤ g(x) for all x in [a, b], this research aims to estab- lish the properties of the Riemann-Stieltjes integral under these integration limits. By extending the integral boundaries to continuous functions, we uncover new properties and insights that further enrich the integral theory landscape. 2. Preliminaries This study focuses on the concept of the limit of operator-valued continuous functions and the Stieltjes limit for operator-valued functions. Therefore, a deep understanding of the Stieltjes limit and continuous functions in the space C[a, b] is required. Definition 1. [13] The set C[a, b] consists of all real-valued continuous functions defined on [a, b] ⊂ R. In defining and proving theorems, a unit function in C[a, b] is required to ensure that the results remain continuous functions. The unit function is defined as follows: Definition 2. [6] The function e is the unit element in C[a, b], where e(x) = 1 for all x in the interval [a, b]. The definition of the Stieltjes limit for function-valued operators involves operations on continuous functions. Therefore, we present some properties of continuous functions in C[a, b] as follows: Theorem 1. [6] For any f, g ∈ C[a, b] and any α, β ∈ R, the following holds: (i) αf + βg ∈ C[a, b] M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 3 of 17 (ii) fg ∈ C[a, b] (iii) f g ∈ C[a, b] if g(x) ̸= 0 for all x ∈ [a, b] In addition to unit function, the ordering properties of continuous functions in C[a, b] are also necessary for defining the Stieltjes limit. The ordering properties of continuous functions in C[a, b] is as follows: Definition 3. [6] For any f and g in C[a, b], we define: (i) f = g ⇔ f(x) = g(x) for every x ∈ [a, b]. (ii) f ⪯ g ⇔ f(x) ≤ g(x) for every x ∈ [a, b]. (iii) f ≺ g ⇔ f(x) < g(x) for every x ∈ [a, b]. (iv) f ⪰ g ⇔ f(x) ≥ g(x) for every x ∈ [a, b]. (v) f ≻ g ⇔ f(x) > g(x) for every x ∈ [a, b]. Theorem 2. [14] For any f, g, and h in C[a, b], the following holds: (i) f ≺ g ⇒ (f + h ≺ g + h) for every h ∈ C[a, b]. (ii) f ≺ g ⇒ (αf ≺ αg) for every α preceded by 0. To compute the Stieltjes limit for function-valued operators, we define open and closed intervals, which serve as the domain for the limit process, as follows: Definition 4. [13] For any f and g in C[a, b] such that f ≺ g, we define: (i) (f, g) = {h ∈ C[a, b] : f ≺ h ≺ g} as the open interval in C[a, b]. (ii) [f, g] = {h ∈ C[a, b] : f ⪯ h ⪯ g} as the closed interval in C[a, b]. Definition 5. [15] Let A ⊆ C[a, b] be a non-empty set. (i) The set A is said to be bounded above if there exists m ∈ C[a, b] such that m ⪰ f for every f ∈ A. In this case, m is called an upper bound of A. (ii) The set A is said to be bounded below if there exists n ∈ C[a, b] such that n ⪯ f for every f ∈ A. In this case, n is called a lower bound of A. (iii) The set A is said to be bounded if it is both bounded above and below. (iv) If A is bounded above, a function u ∈ C[a, b] is called the least upper bound or supremum of A if u is an upper bound of A and for every upper bound m of A, we have u ⪯ m, denoted as: u = sup(A). (v) If A is bounded below, a function l ∈ C[a, b] is called the greatest lower bound or infimum of A if l is a lower bound of A and for every lower bound n of A, we have l ⪰ n, denoted as: l = inf(A). M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 4 of 17 3. Main Theorem This section addresses the limit of operators on C[a, b] and the Stieltjes limit on C[a, b]. The proof of the main theorem is established through a rigorous analysis, following a structured approach that builds upon the results and concepts developed in the preceding subsections, which serve as the theoretical foundation for the argument. 3.1. Limit of Operators on C[a, b] In this subsection, we define the concept of the limit point of a set of continuous func- tions and the limit of an operator whose values are continuous functions. This definition serves as the foundation for defining the Stieltjes limit on operators that take continuous function values. Definition 6. Given f ∈ C[a, b] and a positive real number ε, the ε-neighborhood of the function f is defined as the set: Vε(f) = {g ∈ C[a, b] : |g − f | ≺ εe}. For example, if we consider 0 ∈ C[0, 1], the 3-neighborhood of 0 is given by V3(0) = {g ∈ C[0, 1] : |g − 0| ≺ 3e} = {g ∈ C[0, 1] : |g| ≺ 3e}. Definition 7. Given A ⊂ C[a, b], a function f0 ∈ C[a, b] is called a limit point of the set A if, for any real number ε > 0, the following condition holds: (Vε(f0) ∩A) \ {f0} ≠ ∅. The function f0 may or may not be contained in A. Explicitly, there must exist a function in (Vε(f0) ∩ A) that is different from f0 for f0 to be a limit point of the set A. As an illustration, consider the following example. Example 1. Given A = (g1, g2) ∪ {g3} ⊂ C[1, 2] with g1(x) = x, g2(x) = 3x, and g3(x) = 5x, each defined on the interval [1, 2], we will show that the function h ∈ A with h(x) = 2x and g1 /∈ A is a limit point of A, while g3 ∈ A is not a limit point of A. The function h(t) is a limit point of the set A because for any real number ε > 0, there exists: f(x) = { h+ 1 2εe, if ε < 1, 3 2h, if ε > 1, such that f ∈ (Vε(h) ∩A) and f ̸= h, or equivalently, (Vε(h) ∩A) \ {h} ≠ ∅. Similarly, the function g1 /∈ A is a limit point of the set A because for any real number ε > 0, there exists: f(x) = { g + 1 2εe, if ε < 2, 2g1, if ε > 2, such that f ∈ (Vε(g1) ∩A) and f ̸= g1, or equivalently, (Vε(g1) ∩A) \ {g1} ≠ ∅. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 5 of 17 Next, we show that g3 ∈ A is not a limit point of the set A. The function g3 ∈ A is not a limit point of A because there exists a real number ε = 1 such that (Vε(g3) ∩A) \ {g3} = {g3}, which simplifies to (Vε(g3) ∩A) \ {g3} = ∅. Definition 8. Given A ⊆ C[a, b] and a function f0 as a limit point of the set A, an operator F : A → C[a, b] has a limit operator L at f0 if for every real number ε > 0, there exists a real number δ > 0 such that for every f ∈ A satisfying 0 ≺ |f − f0| ≺ δe, the following holds: |F (f)− L| ≺ εe. The inequality 0 ≺ |f − f0| is equivalent to the statement f ̸= f0. If L is the limit operator of F at f0, then it can be written as: L = lim f→f0 F (f). Example 2. It will be shown that: lim f→2e (2f + e) = 5e. For any positive real number ε, we choose a positive real number δ = ε 2 such that for every f ∈ A satisfying 0 ≺ |f − 2e| ≺ δe, we obtain: |(2f + e)− (5e)| ≺ |2f − 4e| = |2(f − 2e)|. | 2(f − 2e)| = |2| |f − 2e| = 2|f − 2e| ≺ 2δe = 2δe. By substituting δ = ε 2 , we obtain: 2δe = 2 (ε 2 ) e = εe. Thus: lim f→2e (2f + e) = 5e. Example 3. Consider F : C[a, b] → C[a, b] defined as: F (f) = { e, if f ⪯ 0, 2e, if f ≻ 0. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 6 of 17 It will be shown that F (0) = e is not a limit operator of F at 0. The continuous function F (0) = e is not a limit operator of F at 0 because there exists a positive real number ε = 1 2 and any real number δ > 0 such that there exists f0 = (0+ 1 2δe) ∈ A satisfying: 0 ≺ |f0 − 0| ≺ δe, but: |F (f0)− F (0)| = |2e− e| = |e| = e ≻ 1 2 e = εe. Thus, F (0) = e is not a limit operator of F at 0. Before discussing the properties of limit operators on C[a, b], we define the addition, subtraction, multiplication, and division of operators on C[a, b]. This definition is crucial for the subsequent discussion. Definition 9. Given A ⊆ C[a, b] and operators F and G, each mapping A to C[a, b], the definitions of addition F +G, subtraction F −G, and multiplication FG, each from A to C[a, b], are as follows: (F +G)(f) = F (f) +G(f), (F −G)(f) = F (f)−G(f), (FG)(f) = F (f)G(f), For every f ∈ A. In addtition, if γ ∈ C[a, b], we define γF as follows: (γF )(f) = γF (f), For every f ∈ A. Finally, if G(f)(x) ̸= 0 for all x ∈ [a, b], we define F G as follows:( F G ) (f) = F (f) G(f) , for every f ∈ A. Theorem 3. If f0 ∈ C[a, b] such that 0 ⪯ f0 ≺ εe for every ε > 0, then f0 = 0. Proof. Suppose f0 ≻ 0. Choosing a positive real number ε0 < min{1 2f0(x)|x ∈ [a, b]}, we obtain: 0 ≺ ε0e ⪯ 1 2 f0 ≺ f0 or equivalently, ε0e ≺ f0. This contradicts f0 ≺ εe for every positive real number ε. Thus, it must be that f0 = 0. From Definition 4.1.7 and Theorem 4.1.4, several properties of the limit operator on C[a, b] can be established as follows: M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 7 of 17 Theorem 4. Given A ⊆ C[a, b], an operator F mapping from A to C[a, b], and f0 as a limit point of A, if: lim f→f0 F (f) = L and lim f→f0 F (f) = M, then L = M . Proof. Given that limf→f0 F (f) = L, it follows that for every real number ε > 0, there exists δ1 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ1e, then: |F (f)− L| ≺ ε 2 e. Similarly, given limf→f0 F = M , for every ε > 0, there exists δ2 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ2e, then: |F (f)−M | ≺ ε 2 e. Choosing δ = min(δ1, δ2), if 0 ≺ |f − f0| ≺ δe, then: |L−M | = |L− F (f) + F (f)−M | ⪯ |L− F (f)|+ |F (f)−M |. By substitution, we obtain: |L−M | ≺ ε 2 e+ ε 2 e = εe. This holds for every ε > 0, it follows that: |L−M | = 0 or L = M. The implication of Theorem 4 is that if an operator F has a limit at f0, then the limit is unique. Theorem 5. Given A ⊆ C[a, b], operators F and G, each mapping from A to C[a, b], and f0 as a limit point of the set A, along with a function γ ∈ C[a, b]. If limf→f0 F = L and limf→f0 G = M , then: (i) limf→f0(F (f) +G(f)) = L+M (ii) limf→f0(F (f)−G(f)) = L−M (iii) limf→f0(F (f)G(f)) = LM (iv) limf→f0(γF (f)) = γL (v) limf→f0 ( F (f) G(f) ) = L M , provided that G(f)(x) ̸= 0 and M(x) ̸= 0, for all x ∈ [a, b]. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 8 of 17 Proof. (i) Given that limf→f0 F (f) = L, it follows that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ1e, then: |F (f)− L| ≺ ε 2 e. Similarly, given that limf→f0 G(f) = M , it follows that for every real number ε > 0, there exists a real number δ2 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ2e, then: |G(f)−M | ≺ ε 2 e. By choosing δ = min(δ1, δ2), we obtain that for every f ∈ A and 0 ≺ |f − f0| ≺ δe, the following holds: |[F (f) +G(f)]− [L+M ]| ⪯ |F (f)− L|+ |G(f)−M | ≺ ε 2 e+ ε 2 e = εe. Thus, lim f→f0 (F (f) +G(f)) = L+M. (ii) Given that limf→f0 F (f) = L, it follows that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ1e, then: |F (f)− L| ≺ ε 2 e. Similarly, given that limf→f0 G(f) = M , it follows that for every real number ε > 0, there exists a real number δ2 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ2e, then: |G(f)−M | ≺ ε 2 e. By choosing δ = min(δ1, δ2), we obtain that for every f ∈ A and 0 ≺ |f − f0| ≺ δe, the following holds: |[F (f)−G(f)]− [L−M ]| ⪯ |F (f)− L|+ |G(f)−M | ≺ ε 2 e+ ε 2 e = εe. Thus, lim f→f0 (F −G) = L−M. (iii) Given that limf→f0 F (f) = L, it follows that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ1e, then: |F (f)− L| ≺ ε ( e 2|M |+ e ) . M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 9 of 17 Similarly, given that limf→f0 G(f) = M , it follows that for every real number ε > 0, there exists a real number δ2 > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δ2e, then: |G(f)−M | ≺ ε ( e 2(sup |F (f)|+ e) ) , where sup |F (f)| is the supremum of |F (f)| in the neighborhood of f0 where f ̸= f0. By choosing δ = min(δ1, δ2), we obtain that for every 0 ≺ |f−f0| ≺ δe, the following holds: |F (f)G(f)− LM | = |F (f)G(f)− F (f)M + F (f)M − LM | = |F (f)[G(f)−M ] +M [F (f)− L]| ⪯ |F (f)| |G(f)−M |+ |M | |F (f)− L|. From this, we conclude: lim f→f0 (F (f)G(f)) = LM. (iv) Given that limf→f0 F (f) = L, it follows that for every real number ε > 0, there exists a real number δ > 0 such that if f ∈ A and 0 ≺ |f − f0| ≺ δe, then: |F (f)− L| ≺ εe |γ| . Since γ ∈ C[a, b], for every 0 ≺ |f − f0| ≺ δe, the following holds: |γF (f)− γL| = |γ| |F (f)− L| ≺ |γ| εe |γ| = εe. Thus, lim f→f0 (γF (f)) = γL. Next, the concept of a one-sided limit of an operator will be defined in more detail. This definition will be used to understand the behavior of a function around a specific operator in a single direction. The following defines a one-sided limit of an operator on C[a, b]. Definition 10. Consider A ⊆ C[a, b] and let f0 be a limit point of the set A. (i) For an operator F : A → C[a, b], the function L is called the right limit of the operator F at f0 if, for every positive real number ε, there exists a positive real number δ such that for every function f ∈ A ∩ {f ∈ C[a, b] : f ≻ f0} and 0 ≺ |f − f0| ≺ δe, the following holds: |F (f)− F (f0)| ≺ εe. If L is the right limit of the operator F at f0, it is written as L = lim f→f+ 0 F (f). M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 10 of 17 (ii) For an operator F : A → C[a, b], the function L is called the left limit of the operator F at f0 if, for every real number ε > 0, there exists a real number δ > 0 such that for every f ∈ A ∩ {f ∈ C[a, b] : f ≺ f0} and 0 ≺ |f − f0| ≺ δe, the following holds: |F (f)− F (f0)| ≺ εe. If L is the left limit of the operator F at f0, it is written as L = lim f→f− 0 F (f). Example 4. Consider F : C[a, b] → C[a, b] defined by: F (f) = { e, if f ⪯ 0, 2e, if f ≻ 0. We will show that lim f→0+ F (f) = 2e and lim f→0− F (f) = e. (i) Take any positive real number ε, and choose a positive real number δ = 5 such that for every f ∈ C[a, b] ∩ {f ∈ C[a, b] : f ≻ 0} and 0 ≺ |f − 0| ≺ δe, We obtain: |F (f)− 2e| = |2e− 2e| = |0| = 0 < εe. Thus: lim f→0+ F (f) = 2e. (ii) Take any positive real number ε, and choose a positive real number δ = 5 such that for every f ∈ C[a, b] ∩ {f ∈ C[a, b] : f ≺ 0} and 0 ≺ |f − 0| ≺ δe, We obtain: |F (f)− e| = |e− e| = |0| = 0 < εe. Thus: lim f→0− F (f) = e. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 11 of 17 Next, we discuss a theorem that establishes the relationship between the general defi- nition of the limit of an operator and the definition of the one-sided limit of an operator (right-hand limit or left-hand limit), as described below. Theorem 6. Consider F : A ⊆ C[a, b] → C[a, b] and let f0 ∈ A be a limit point of the set A ∩ {f ∈ C[a, b] : f ≻ f0} and the set A ∩ {f ∈ C[a, b] : f ≺ f0}. We obtain limf→f0 F (f) = L if and only if lim f→f+ 0 F (f) = L = lim f→f− 0 F (f). Proof. Consider F : A ⊆ C[a, b] → C[a, b] and let f0 ∈ A be a limit point of the set A ∩ {f ∈ C[a, b] : f ≻ f0} and the set A ∩ {f ∈ C[a, b] : f ≺ f0}. (⇒) Suppose limf→f0 F = L, meaning that for every real number ε > 0, there exists a real number δ > 0 such that for every f ∈ A and 0 ≺ |f − f0| ≺ δe, we have: |F (f)− L| ≺ εe. Since A ∩ {f ∈ C[a, b] : f ≻ f0} ⊆ A and A ∩ {f ∈ C[a, b] : f ≺ f0} ⊆ A, it follows that for every real number ε > 0, there exists a real number δ > 0 such that for every f ∈ A ∩ {f ∈ C[a, b] : f ≻ f0} and 0 ≺ |f − f0| ≺ δe, we obtain: |F (f)− L| ≺ εe. Similarly, for every f ∈ A ∩ {f ∈ C[a, b] : f ≺ f0} and 0 ≺ |f − f0| ≺ δe, we have: |F (f)− L| ≺ εe. Thus, we conclude: lim f→f+ 0 F (f) = L and lim f→f− 0 F (f) = L. (⇐) Suppose limf→f+ 0 F (f) = L, meaning that for every real number ε > 0, there exists a real number δ1 > 0 such that for every f ∈ A ∩ {f ∈ C[a, b] : f ≻ f0} and 0 ≺ |f − f0| ≺ δ1e, we obtain: |F (f)− L| ≺ εe. Similarly, since limf→f− 0 F (f) = L, for every real number ε > 0, there exists a real number δ2 > 0 such that for every f ∈ A∩{f ∈ C[a, b] : f ≺ f0} and 0 ≺ |f −f0| ≺ δ2e, we obtain: |F (f)− L| ≺ εe. Choosing δ = min{δ1, δ2}, we conclude that for every real number ε > 0, there exists a real number δ > 0 such that for every f ∈ A and 0 ≺ |f − f0| ≺ δe, the following holds: |F (f)− L| < εe. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 12 of 17 Thus, lim f→f0 F (f) = L. Next, we define the concept of operator continuity at f0 ∈ A ⊆ C[a, b] or over B ⊆ C[a, b], which is frequently used in subsequent subsections. Definition 11. Given A ⊆ C[a, b] and an operator R : A → C[a, b], the operator R is said to be continuous at f0 if, for every real number ε > 0, there exists δ > 0 such that for every f ∈ A with |f − f0| ≺ δe, the following holds: |R(f)−R(f0)| ≺ εe. Furthermore, the operator R is said to be continuous on the set B if R is continuous at every f ∈ B. 3.2. Stieltjes Limit on C[a, b] Before defining the Stieltjes limit on C[a, b], it is important to first establish the defini- tion of increasing, strictly increasing, decreasing, and strictly decreasing operators. These fundamental concepts serve as a crucial foundation for understanding the definition and properties of the Stieltjes limit on C[a, b]. The following definitions describe increasing, strictly increasing, decreasing, and strictly decreasing operators: Definition 12. (i) An operator R is said to be increasing on A if for every t, v ∈ A with t ≺ v, then R(t) ⪯ R(v). (ii) An operator R is said to be strictly increasing on A if for every t, v ∈ A with t ≺ v, then R(t) ≺ R(v). (iii) An operator R is said to be decreasing on A if for every t, v ∈ A with t ≺ v, then R(t) ⪰ R(v). (iv) An operator R is said to be strictly decreasing on A if for every t, v ∈ A with t ≺ v, then R(t) ≻ R(v). Next, we define and discuss the Stieltjes limit Theorem for an operator R in the space of continuous functions C[a, b]. The Stieltjes limit of an operator with continuous function values generalizes the concept of the limit of an operator with continuous function values. Definition 13. Given A ⊆ C[a, b] and a function f0 as a limit point of the set A, an operator F : A → C[a, b], and an operator R : A → C[a, b] that is continuous and increasing, a function L is called the limit operator of F at f0 with respect to R if, for every real number ε > 0, there exists a real number δ > 0 such that for every f ∈ A satisfying 0 ≺ |R(f)−R(f0)| ≺ δe, the following holds: |F (f)− L| ≺ εe. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 13 of 17 If L is the limit operator of F with respect to R at f0, it is written as: L = lim R(f)→R(f0) F (f). Next, in this subsection, each operator F and R considered is an operator mapping from A ⊆ C[a, b] to C[a, b] that is defined on an open interval, with R being a continuous and increasing operator. Example 5. Given R(f) = 2f and F (f) = 2f + e, we will show that: lim R(f)→R(e) (2f + e) = 3e. Taking any positive real number ε, we choose a positive real number δ = ε such that for every f ∈ A satisfying 0 ≺ |R(f)−R(e)| = |2f − 2e| ≺ δe, we obtain: |(2f + e)− (3e)| ≺ |2f − 2e| ≺ δe = εe. Thus: lim f→2e (2f + e) = 3e. Theorem 7. Given A ⊆ C[a, b], an operator F mapping from A to C[a, b], and f0 as a limit point of the set A, if: lim R(f)→R(f0) F (f) = L and lim R(f)→R(f0) F (f) = M, then L = M . Proof. Given limR(f)→R(f0) F (f) = L, meaning that for every real number ε > 0, there exists δ1 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ1e, then: |F (f)− L| ≺ ε 2 e. Similarly, given limR(f)→R(f0) F (f) = M , meaning that for every ε > 0, there exists δ2 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ2e, then: |F (f)−M | ≺ ε 2 e. Choosing δ = min(δ1, δ2), if 0 ≺ |f − f0| ≺ δe, then: |L−M | = |L− F (f) + F (f)−M | ⪯ |L− F (f)|+ |F (f)−M |. By substitution, we obtain: |L−M | ≺ ε 2 e+ ε 2 e = εe. This holds for every ε > 0, by Theorem 4.1.11: |L−M | = 0 or L = M. The significance of Theorem 4.5.4 is that if an operator F has a limit at f0 with respect to R, then the limit is unique. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 14 of 17 Theorem 8. Given A ⊆ C[a, b], operators F , G, and R, each mapping from A to C[a, b], with R being an increasing operator, and f0 as a limit point of the set A, along with a function γ ∈ C[a, b]. If limR(f)→R(f0) F (f) = L and limR(f)→R(f0)G(f) = M , then: (i) limR(f)→R(f0)(F +G)(f) = L+M (ii) limR(f)→R(f0)(F −G) = L−M (iii) limR(f)→R(f0)(FG)(f) = LM (iv) limR(f)→R(f0)(γF (f)) = γL (v) limR(f)→R(f0) ( F G ) (f) = L M , provided that M(x) ̸= 0 for all x ∈ [a, b]. Proof. (i) Given that limR(f)→R(f0) F = L, meaning that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ1e, then: |F (f)− L| ≺ ε 2 e. Similarly, given that limR(f)→R(f0)G = M , meaning that for every real number ε > 0, there exists a real number δ2 > 0 such that if f ∈ A and 0 ≺ |R(f)− R(f0)| ≺ δ2e, then: |G(f)−M | ≺ ε 2 e. Choosing δ = min(δ1, δ2), we obtain that for every f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δe, the following holds: |[F (f) +G(f)]− [L+M ]| ⪯ |F (f)− L|+ |G(f)−M | ≺ ε 2 e+ ε 2 e = εe. Thus, lim R(f)→R(f0) (F +G)(f) = L+M. (ii) Given that limR(f)→R(f0) F (f) = L, meaning that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ1e, then: |F (f)− L| ≺ ε 2 e. Similarly, given that limR(f)→R(f0)G(f) = M , meaning that for every real number ε > 0, there exists a real number δ2 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ2e, then: |G(f)−M | ≺ ε 2 e. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 15 of 17 Choosing δ = min(δ1, δ2), we obtain that for every f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δe, the following holds: |[F (f)−G(f)]− [L−M ]| ⪯ |F (f)− L|+ |G(f)−M | ≺ ε 2 e+ ε 2 e = εe. Thus, lim R(f)→R(f0) (F −G)(f) = L−M. (iii) Given that limR(f)→R(f0) F (f) = L, meaning that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ1e, then: |F (f)− L| ≺ ε ( e 2|M |+ e ) . Similarly, given that limR(f)→R(f0)G(f) = M , meaning that for every real number ε > 0, there exists a real number δ2 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ2e, then: |G(f)−M | ≺ ε ( e 2(sup |F (f)|+ e) ) , where sup |F (f)| is the supremum of |F (f)| in the neighborhood of f0 where f ̸= f0. Choosing δ = min(δ1, δ2), we obtain that for every 0 ≺ |R(f) − R(f0)| ≺ δe, the following holds: |F (f)G(f)− LM | = |F (f)G(f)− F (f)M + F (f)M − LM | = |F (f)[G(f)−M ] +M [F (f)− L]| ⪯ |F (f)| |G(f)−M |+ |M | |F (f)− L|. Thus, lim R(f)→R(f0) (FG)(f) = LM. (iv) Given that limR(f)→R(f0) F (f) = L, meaning that for every real number ε > 0, there exists a real number δ > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δe, then: |γF (f)− γL| = |γ| |F (f)− L| ≺ |γ| εe |γ| = εe. Thus, lim R(f)→R(f0) (γF (f)) = γL. M. Alifuddin et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5909 16 of 17 (v) Given that limR(f)→R(f0) F (f) = L, meaning that for every real number ε > 0, there exists a real number δ1 > 0 such that if f ∈ A and 0 ≺ |R(f)−R(f0)| ≺ δ1e, then: |F (f)− L| ≺ |M | 4 e. Similarly, given limR(f)→R(f0)G = M , the same holds, ensuring: lim R(f)→R(f0) ( F G ) (f) = L M . 4. Conclusion The findings of this study indicate that the characteristics of the limit and the Stieltjes limit of function-valued continuous functions are comparable to those of the limit and the Stieltjes limit of real-valued functions. Acknowledgements The author(s) would like to thank the Department of Mathematics, Institut Teknologi Sepuluh Nopember (ITS), Surabaya, Indonesia, for supporting this research. Additionally, the author(s) express their gratitude to Lembaga Pengelola Dana Pendidikan (LPDP) and Balai Pembiayaan Pendidikan Tinggi (BPPT) Indonesia for their financial support of this research. References [1] Umi Mahnuna Hanung and Ch Rini Indrati. 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