# Ideal quasi-Cauchy sequences

- Huseyin Cakalli
^{1}Email author and - Bipan Hazarika
^{2}

**2012**:234

https://doi.org/10.1186/1029-242X-2012-234

© Cakalli and Hazarika; licensee Springer 2012

**Received: **20 July 2012

**Accepted: **28 September 2012

**Published: **17 October 2012

## Abstract

An ideal *I* is a family of subsets of positive integers **N** which is closed under taking finite unions and subsets of its elements. A sequence $({x}_{n})$ of real numbers is said to be *I*-convergent to a real number *L* if for each $\epsilon >0$, the set $\{n:|{x}_{n}-L|\ge \epsilon \}$ belongs to *I*. We introduce *I*-ward compactness of a subset of **R**, the set of real numbers, and *I*-ward continuity of a real function in the senses that a subset *E* of **R** is *I*-ward compact if any sequence $({x}_{n})$ of points in *E* has an *I*-quasi-Cauchy subsequence, and a real function is *I*-ward continuous if it preserves *I*-quasi-Cauchy sequences where a sequence $({x}_{n})$ is called to be *I*-quasi-Cauchy when $(\mathrm{\Delta}{x}_{n})$ is *I*-convergent to 0. We obtain results related to *I*-ward continuity, *I*-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, *δ*-ward continuity, and slowly oscillating continuity.

**MSC:** 40A35, 40A05, 40G15, 26A15.

## Keywords

## 1 Introduction

The concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences involving mathematics, especially in computer science, information theory, and biological science.

A subset *E* of **R**, the set of real numbers, is compact if any open covering of *E* has a finite subcovering where **R** is the set of real numbers. This is equivalent to the statement that any sequence $\mathbf{x}=({x}_{n})$ of points in *E* has a convergent subsequence whose limit is in *E*. A real function *f* is continuous if and only if $(f({x}_{n}))$ is a convergent sequence whenever $({x}_{n})$ is. Regardless of limit, this is equivalent to the statement that $(f({x}_{n}))$ is Cauchy whenever $({x}_{n})$ is. Using the idea of continuity of a real function and the idea of compactness in terms of sequences, Çakallı [1] introduced the concept of ward continuity in the sense that a function *f* is ward continuous if it preserves quasi-Cauchyness, *i.e.*, $(f({x}_{n}))$ is a quasi-Cauchy sequence whenever $({x}_{n})$ is, and a subset *E* of **R** is ward compact if any sequence $\mathbf{x}=({x}_{n})$ of points in *E* has a quasi-Cauchy subsequence $\mathbf{z}=({z}_{k})=({x}_{{n}_{k}})$ of the sequence **x** where a sequence $({z}_{k})$ is called quasi-Cauchy if ${lim}_{k\to \mathrm{\infty}}({z}_{k+1}-{z}_{k})=0$ (see also [2], [3] and [4]). In [5] a real-valued function defined on a subset *E* of **R** is called *δ*-ward continuous if it preserves *δ*-quasi Cauchy sequences where a sequence $\mathbf{x}=({x}_{n})$ is defined to be *δ*-quasi Cauchy if the sequence $(\mathrm{\Delta}{x}_{n})$ is quasi-Cauchy. A subset *E* of **R** is said to be *δ*-ward compact if any sequence $\mathbf{x}=({x}_{n})$ of points in *E* has a *δ*-quasi-Cauchy subsequence $\mathbf{z}=({z}_{k})=({x}_{{n}_{k}})$ of the sequence **x**.

A sequence $({x}_{n})$ of points in **R** is slowly oscillating if for any given $\epsilon >0$, there exists a $\delta =\delta (\epsilon )>0$ and an $N=N(\epsilon )$ such that $|{x}_{m}-{x}_{n}|<\epsilon $ if $n\ge N(\epsilon )$ and $n\le m\le (1+\delta )n$. A function defined on a subset *E* of **R** is called slowly oscillating continuous if it preserves slowly oscillating sequences (see [6]). A function defined on a subset *E* of **R** is called quasi-slowly oscillating continuous on *E* if it preserves quasi-slowly oscillating sequences of points in *E* where a sequence $\mathbf{x}=({x}_{n})$ is called quasi-slowly oscillating if $(\mathrm{\Delta}{x}_{n})=({x}_{n+1}-{x}_{n})$ is a slowly oscillating sequence [7].

The purpose of this paper is to investigate the concept of ideal ward compactness of a subset of **R** and the concept of ideal ward continuity of a real function which cannot be given by means of any summability matrix and to prove related theorems.

## 2 Preliminaries

First of all, some definitions and notation will be given in the following. Throughout this paper, **N** and **R** will denote the set of all positive integers and the set of all real numbers, respectively. We will use boldface letters
,
,
, … for sequences $\mathbf{x}=({x}_{n})$, $\mathbf{y}=({y}_{n})$, $\mathbf{z}=({z}_{n})$, … of terms in **R**. *c* and *S* will denote the set of all convergent sequences and the set of all statistically convergent sequences of points in **R**, respectively.

Following the idea given in a 1946 American Mathematical Monthly problem [8], a number of authors such as Posner [9], Iwinski [10], Srinivasan [11], Antoni [12], Antoni and Salat [13], Spigel and Krupnik [14] have studied *A*-continuity defined by a regular summability matrix *A*. Some authors, Öztürk [15], Savaş and Das [16], Borsik and Salat [17], have studied *A*-continuity for methods of almost convergence or for related methods.

*et al.*The concept of statistical convergence is a generalization of the usual notion of convergence that, for real-valued sequences, parallels the usual theory of convergence. For a subset

*M*of

**N**, the asymptotic density of

*M*denoted by $\delta (M)$ is given by

*ℓ*if

for every $\u03f5>0$. In this case, *ℓ* is called the statistical limit of **x**. Schoenberg [20] studied some basic properties of statistical convergence and also studied the statistical convergence as a summability method. Fridy [34] gave characterizations of statistical convergence. Caserta, Maio, and Kočinac [35] studied statistical convergence in function spaces, while Caserta and Kočinac [36] investigated statistical exhaustiveness.

*θ*will be denoted by ${I}_{r}=({k}_{r-1},{k}_{r}]$, and the ratio $\frac{{k}_{r}}{{k}_{r-1}}$ will be abbreviated as ${q}_{r}$. In this paper, we assume that ${lim\hspace{0.17em}inf}_{r}{q}_{r}>1$. The notion of lacunary statistical convergence was introduced and studied by Fridy and Orhan in [37] and [38] (see also [39] and [40]). A sequence $({\alpha}_{k})$ of points in

**R**is called lacunary statistically convergent to an element

*ℓ*of

**R**if

for every positive real number *ε*. The condition assumed a few lines above ensures the regularity of the lacunary statistical sequential method.

The concept of *I*-convergence, which is a generalization of statistical convergence, was introduced by Kostyrko, S̆alàt, and Wilczyński [41] by using the ideal *I* of subsets of **N** and further studied in [42–46], and [47]. The concept was also studied for double sequences in [48–51], and [52]. Although an ideal is defined as a hereditary and additive family of subsets of a non-empty arbitrary set *X*, here in our study, it suffices to take *I* as a family of subsets of **N** such that $A\cup B\in I$ for each $A,B\in I$, and each subset of an element of *I* is an element of *I*. A non-trivial ideal *I* is called *admissible* if and only if $\{\{n\}:n\in \mathbf{N}\}\subset I$. A non-trivial ideal *I* is maximal if there cannot exist any non-trivial ideal $J\ne I$ containing *I* as a subset. Further details on ideals can be found in Kostyrko *et al.* (see [41]). Throughout this paper, we assume *I* is a non-trivial admissible ideal in **N**. Recall that a sequence
of points in **R** is said to be *I*-convergent to a real number *ℓ* if
for every $\epsilon >0$. In this case, we write $I\text{-}lim{x}_{n}=\ell $. We say that a sequence $({x}_{n})$ of points in **R** is *I*-quasi-Cauchy if $I\text{-}lim({x}_{n+1}-{x}_{n})=0$. We see that *I*-convergence of a sequence $({x}_{n})$ implies *I*-quasi-Cauchyness of $({x}_{n})$. Throughout the paper, $I(\mathbf{R})$ and Δ*I* will denote the set of all *I*-convergent sequences and the set of all *I*-quasi-Cauchy sequences of points in **R**, respectively.

If we take $I={I}_{\mathrm{fin}}=\{A\subseteq \mathbf{N}:A\text{is a finite subset of}\mathbf{N}\}$, then ${I}_{\mathrm{fin}}$ will be a non-trivial admissible ideal in $\mathbf{N},$ and the corresponding convergence will coincide with the usual convergence. If we take $I={I}_{\delta}=\{A\subseteq \mathbf{N}:\delta (A)=0\}$, then ${I}_{\delta}$ will be a non-trivial admissible ideal of **N**, and the corresponding convergence will coincide with the statistical convergence.

Connor and Grosse-Erdman [53] gave sequential definitions of continuity for real functions calling *G*-continuity instead of *A*-continuity. Their results cover the earlier works related to *A*-continuity where a method of sequential convergence, or briefly a method, is a linear function *G* defined on a linear subspace of *s*, denoted by ${c}_{G}$, into **R**. A sequence $\mathbf{x}=({x}_{n})$ is said to be *G*-convergent to *ℓ* if $\mathbf{x}\in {c}_{G}$ and $G(\mathbf{x})=\ell $. In particular, lim denotes the limit function $lim\mathbf{x}={lim}_{n}{x}_{n}$ on the linear space *c*, and $\mathit{st}\text{-}lim$ denotes the statistical limit function $\mathit{st}\text{-}lim\mathbf{x}=\mathit{st}\text{-}{lim}_{n}{x}_{n}$ on the linear space *S*. Also $I\text{-}lim$ denotes the *I*-limit function $I\text{-}lim\mathbf{x}=I\text{-}{lim}_{n}{x}_{n}$ on the linear space $I(\mathbf{R})$. A method *G* is called regular if every convergent sequence $\mathbf{x}=({x}_{n})$ is *G*-convergent with $G(\mathbf{x})=lim\mathbf{x}$. A method is called subsequential if whenever **x** is *G*-convergent with $G(\mathbf{x})=\ell $, then there is a subsequence $({x}_{{n}_{k}})$ of **x** with ${lim}_{k}{x}_{{n}_{k}}=\ell $. Since the ordinary convergence implies ideal convergence, *I* is a regular sequential method [54]. Recently, Cakalli studied new sequential definitions of compactness in [55, 56] and slowly oscillating compactness in [6].

## 3 Ideal sequential compactness

First, we recall the definition of *G*-sequentially compactness of a subset *E* of **R**. A subset *E* of **R** is called *G*-sequentially compact if whenever $({x}_{n})$ is a sequence of points in *E*, there is subsequence $\mathbf{y}=({y}_{k})=({x}_{{n}_{k}})$ of $({x}_{n})$ such that $G(\mathbf{y})=lim\mathbf{y}$ in *E* (see [57]). For regular methods, any sequentially compact subset *E* of **R** is also *G*-sequentially compact and the converse is not always true. For any regular subsequential method *G*, a subset *E* of **R** is *G*-sequentially compact if and only if it is sequentially compact in the ordinary sense.

Although *I*-sequential compactness is a special case of *G*-sequential compactness when $G=lim$, we state the definition of *I*-sequential compactness of a subset *E* of **R** as follows.

**Definition 1** A subset *E* of **R** is called *I*-sequentially compact if whenever $({x}_{n})$ is a sequence of points in *E*, there is an *I*-convergent subsequence $\mathbf{y}=({y}_{k})=({x}_{{n}_{k}})$ of $({x}_{n})$ such that $I\text{-}lim\mathbf{y}$ is in *E*.

**Lemma 1** [41]

*Sequential method* *I* *is regular*, *i*.*e*., *if* $lim{x}_{n}=\ell $, *then* $I\text{-}lim{x}_{n}=\ell $.

**Lemma 2** ([58], Proposition 3.2)

*Any* *I*-*convergent sequence of points in* **R** *with an* *I*-*limit* *ℓ* *has a convergent subsequence with the same limit* *ℓ* *in the ordinary sense*.

We have the following result which states that any non-trivial admissible ideal *I* is a regular subsequential sequential method.

**Theorem 1** *The sequential method* *I* *is regular and subsequential*.

*Proof* Regularity of *I* follows from Lemma 1, and subsequentiality of *I* follows from Lemma 2. □

**Theorem 2** ([55], Corollary 3)

*A subset of* **R** *is sequentially compact if and only if it is* *I*-*sequentially compact*.

Although *I*-sequential continuity is a special case of *G*-sequential continuity when $G=lim$, we state the definition of *I*-sequential continuity of a function defined on a subset *E* of **R** as follows.

**Definition 2** ([54], Definition 2)

A function $f:E\to \mathbf{R}$ is *I*-sequentially continuous at a point ${x}_{0}$ if, given a sequence $({x}_{n})$ of points in *E*, $I\text{-}lim\mathbf{x}={x}_{0}$ implies that $I\text{-}limf(\mathbf{x})=f({x}_{0})$.

**Theorem 3** *Any* *I*-*sequentially continuous function at a point* ${x}_{0}$ *is continuous at* ${x}_{0}$ *in the ordinary sense*.

*Proof* Although there is a proof in [54], we give a different proof for completeness. Let *f* be any *I*-sequentially continuous function at a point ${x}_{0}$. Since any proper admissible ideal is a regular subsequential method by Theorem 1, it follows from [57], Theorem 13, that *f* is continuous in the ordinary sense. □

**Theorem 4** ([54], Theorem 2.2)

*Any continuous function at a point* ${x}_{0}$ *is* *I*-*sequentially continuous at* ${x}_{0}$.

Combining Theorem 3 and Theorem 4 we have the following corollary.

**Corollary 1** *A function is* *I*-*sequentially continuous at a point* ${x}_{0}$ *if and only if it is continuous at* ${x}_{0}$.

As statistical limit is an *I*-sequential method, we get ([56], Theorem 2).

**Corollary 2** *A function is statistically continuous at a point* ${x}_{0}$ *if and only if it is continuous at* ${x}_{0}$ *in the ordinary sense*.

As lacunary statistical limit is an *I*-sequential method, we get ([56], Theorem 5).

**Corollary 3** *A function is lacunarily statistically continuous at a point* ${x}_{0}$ *if and only if it is continuous at* ${x}_{0}$ *in the ordinary sense*.

**Corollary 4** *For any regular subsequential method* *G*, *a function is* *G*-*sequentially continuous at a point* ${x}_{0}$, *then it is* *I*-*sequentially continuous at* ${x}_{0}$.

*Proof* The proof follows from [57], Theorem 13. □

**Corollary 5** *Any ward continuous function on a subset* *E* *of* **R** *is* *I*-*sequentially continuous on* *E*.

**Theorem 5** *If a function is slowly oscillating continuous on a subset* *E* *of* **R**, *then it is* *I*-*sequentially continuous on* *E*.

*Proof* Let *f* be any slowly oscillating continuous function on *E*. It follows from [6], Theorem 2.1, that *f* is continuous. By Theorem 4, we see that *f* is *I*-sequentially continuous on *E*. This completes the proof. □

**Theorem 6** *If a function is* *δ*-*ward continuous on a subset* *E* *of* **R**, *then it is* *I*-*sequentially continuous on* *E*.

*Proof* Let *f* be any *δ*-ward continuous function on *E*. It follows from [5], Corollary 2, that *f* is continuous. By Theorem 4, we obtain that *f* is *I*-sequentially continuous on *E*. This completes the proof. □

**Corollary 6** *If a function is quasi*-*slowly oscillating continuous on a subset* *E* *of* **R**, *then it is* *I*-*sequentially continuous on* *E*.

*Proof* Let *f* be any quasi-slowly oscillating continuous function on *E*. It follows from [7], Theorem 3.2, that *f* is continuous. By Theorem 4, we deduce that *f* is *I*-sequentially continuous on *E*. This completes the proof. □

## 4 Ideal ward continuity

We say that a sequence $\mathbf{x}=({x}_{n})$ is *I*-ward convergent to a number *ℓ* if $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}{x}_{n}=\ell $. For the special case $\ell =0$, we say that **x** is ideal quasi-Cauchy or *I*-quasi-Cauchy.

Now, we give the definition of *I*-ward compactness of a subset of **R**.

**Definition 3** A subset *E* of **R** is called *I*-ward compact if whenever $\mathbf{x}=({x}_{n})$ is a sequence of points in *E*, there is an *I*-quasi-Cauchy subsequence of **x**.

*I*-ward compactness cannot be obtained by any

*G*-sequential compactness,

*i.e.*, by any summability matrix

*A*, even by the summability matrix $A=({a}_{nk})$ defined by ${a}_{nk}=-1$ if $k=n$ and ${a}_{kn}=1$ if $k=n+1$ and

Despite the fact that *G*-sequential compact subsets of **R** should include the singleton set $\{0\}$, *I*-ward compact subsets of **R** do not have to include the singleton $\{0\}$.

Firstly, we note that any finite subset of **R** is *I*-ward compact, the union of two *I*-ward compact subsets of **R** is *I*-ward compact, and the intersection of any *I*-ward compact subsets of **R** is *I*-ward compact. Furthermore, any subset of an *I*-ward compact set is *I*-ward compact, and any bounded subset of **R** is *I*-ward compact. Any compact subset of **R** is also *I*-ward compact, and the set **N** is not *I*-ward compact. We note that any slowly oscillating compact subset of **R** is *I*-ward compact (see [6] for the definition of slowly oscillating compactness). These observations suggest that we have the following result.

**Theorem 7** *A subset* *E* *of* **R** *is ward compact if and only if it is* *I*-*ward compact*.

*Proof* Let us suppose first that *E* is ward compact. It follows from [2], Lemma 2, that *E* is bounded. Then for any sequence $({x}_{n})$, there exists a convergent subsequence $({x}_{{n}_{k}})$ of $({x}_{n})$ whose limit may be in *E* or not. Then the sequence $(\mathrm{\Delta}{x}_{{n}_{k}})$ is a null sequence. Since *I* is a regular method, $(\mathrm{\Delta}{x}_{{n}_{k}})$ is *I*-convergent to 0, so it is *I*-quasi-Cauchy. Thus, *E* is *I*-ward compact.

Now, to prove the converse, suppose that *E* is *I*-ward compact. Take any sequence $({x}_{n})$ of points in *E*. Then there exists an *I*-quasi-Cauchy subsequence $({x}_{{n}_{k}})$ of $({x}_{n})$. Since *I* is subsequential, there exists a convergent subsequence $({x}_{{n}_{{k}_{m}}})$ of $({x}_{{n}_{k}})$. The sequence $({x}_{{n}_{{k}_{m}}})$ is a quasi-Cauchy subsequence of the sequence $({x}_{n})$. Thus, *E* is ward compact. This completes the proof of the theorem. □

**Theorem 8** *A subset* *E* *of* **R** *is bounded if and only if it is* *I*-*ward compact*.

*Proof* Using an idea in the proof of [2], Lemma 2, and the preceding theorem, the proof can be obtained easily, so it is omitted. □

Now, we give the definition of *I*-ward continuity of a real function.

**Definition 4** A function *f* is called *I*-ward continuous on a subset *E* of **R** if $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}f({x}_{n})=0$ whenever $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}{x}_{n}=0$ for a sequence $\mathbf{x}=({x}_{n})$ of terms in *E*.

We note that this definition of continuity cannot be obtained by any *A*-continuity, *i.e.*, by any summability matrix *A*, even by the summability matrix $A=({a}_{nk})$ defined by (1). However, for this special summability matrix *A*, if *A*-continuity of *f* at the point 0 implies *I*-ward continuity of *f*, then $f(0)=0$; and if *I*-ward continuity of *f* implies *A*-continuity of *f* at the point 0, then $f(0)=0$.

We note that the sum of two *I*-ward continuous functions is *I*-ward continuous, but the product of two *I*-ward continuous functions need not be *I*-ward continuous as it can be seen by considering a product of the *I*-ward continuous function $f(x)=x$ with itself.

In connection with *I*-quasi-Cauchy sequences and *I*-convergent sequences, the problem arises to investigate the following types of ‘continuity’ of functions on **R**:

(*δi*): $({x}_{n})\in \mathrm{\Delta}I\Rightarrow (f({x}_{n}))\in \mathrm{\Delta}I$,

($\delta ic$): $({x}_{n})\in \mathrm{\Delta}I\Rightarrow (f({x}_{n}))\in c$,

(*c*): $({x}_{n})\in c\Rightarrow (f({x}_{n}))\in c$,

($c\delta i$): $({x}_{n})\in c\Rightarrow (f({x}_{n}))\in \mathrm{\Delta}I$,

(*i*): $({x}_{n})\in I(\mathbf{R})\Rightarrow (f({x}_{n}))\in I(\mathbf{R})$.

We see that (*δi*) is *I*-ward continuity of *f*, (*i*) is *I*-continuity of *f* and (*c*) states the ordinary continuity of *f*. It is easy to see that ($\delta ic$) implies (*δi*) and (*δi*) does not imply ($\delta ic$); (*δi*) implies ($c\delta i$) and ($c\delta i$) does not imply (*δi*); ($\delta ic$) implies (*c*) and (*c*) does not imply ($\delta ic$); and (*c*) is equivalent to ($c\delta i$).

Now, we give the implication (*δi*) implies (*i*), *i.e.*, any *I*-ward continuous function is *I*-sequentially continuous.

**Theorem 9** *If* *f* *is* *I*-*ward continuous on a subset* *E* *of* **R**, *then it is* *I*-*sequentially continuous on* *E*.

*Proof*Although the following proof is similar to that of [59], Theorem 1, and that of [7], Theorem 3.2, we give it for completeness. Suppose that

*f*is an

*I*-ward continuous function on a subset

*E*of

**R**. Let $({x}_{n})$ be an

*I*-quasi-Cauchy sequence of points in

*E*. Then the sequence

*I*-quasi-Cauchy sequence. Since

*f*is

*I*-ward continuous, the sequence

is an *I*-quasi-Cauchy sequence. Therefore, $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}{y}_{n}=0$. Hence, $I\text{-}{lim}_{n\to \mathrm{\infty}}[f({x}_{n})-f({x}_{0})]=0$. It follows that the sequence $(f({x}_{n}))$ *I*-converges to $f({x}_{0})$. This completes the proof of the theorem.

The converse is not always true for the function $f(x)={x}^{2}$ is an example since $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}{x}_{n}=0$ for the sequence $({x}_{n})=(\sqrt{n})$. But $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}f({x}_{n})\ne 0$, because $(f(\sqrt{n}))=(n)$. □

**Theorem 10** *If* *f* *is* *I*-*ward continuous on a subset* *E* *of* **R**, *then it is continuous on* *E* *in the ordinary sense*.

*Proof* Let *f* be an *I*-ward continuous function on *E*. By Theorem 9, *f* is *I*-sequentially continuous on *E*. It follows from Theorem 3 that *f* is continuous on *E* in the ordinary sense. Thus, the proof is completed. □

**Theorem 11** *An* *I*-*ward continuous image of any* *I*-*ward compact subset of* **R** *is* *I*-*ward compact*.

*Proof* Suppose that *f* is an *I*-ward continuous function on a subset *E* of **R**, and *E* is an *I*-ward compact subset of **R**. Let $\mathbf{y}=({y}_{n})$ be a sequence of points in $f(E)$. Write ${y}_{n}=f({x}_{n})$, where ${x}_{n}\in E$ for each $n\in \mathbf{N}$. *I*-ward compactness of *E* implies that there is an *I*-quasi-Cauchy subsequence $\mathbf{z}=({z}_{k})=({x}_{{n}_{k}})$ of $({x}_{n})$. As *f* is *I*-ward continuous, $(f({z}_{k}))$ is an *I*-quasi-Cauchy subsequence of **y**. Thus, $f(E)$ is *I*-ward compact. This completes the proof of the theorem. □

**Corollary 7** *An* *I*-*ward continuous image of any compact subset of* **R** *is compact*.

*Proof* The proof of this theorem follows from Theorem 3. □

**Corollary 8** *An* *I*-*ward continuous image of any bounded subset of* **R** *is bounded*.

*Proof* The proof follows from Theorem 8 and Theorem 10. □

**Corollary 9** *An* *I*-*ward continuous image of a* *G*-*sequentially compact subset of* **R** *is* *G*-*sequentially compact for any regular subsequential method* *G*.

It is a well-known result that a uniform limit of a sequence of continuous functions is continuous. It is true for slowly oscillating continuous functions ([56], Theorem 12), and quasi-slowly oscillating continuous functions ([7], Theorem 3.5), (see also [59], Theorem 5). This is also true in the case of *I*-ward continuity, *i.e.*, a uniform limit of a sequence of *I*-ward continuous functions is *I*-ward continuous.

**Theorem 12** *If* $({f}_{n})$ *is a sequence of* *I*-*ward continuous functions defined on a subset* *E* *of* **R**, *and* $({f}_{n})$ *is uniformly convergent to a function* *f*, *then* *f* *is* *I*-*ward continuous on* *E*.

*Proof*Let $\epsilon >0$ and $({x}_{n})$ be a sequence of points in

*E*such that $I\text{-}{lim}_{n\to \mathrm{\infty}}\mathrm{\Delta}{x}_{n}=0$. By the uniform convergence of $({f}_{n})$, there exists a positive integer

*N*such that $|{f}_{n}(x)-f(x)|<\frac{\epsilon}{3}$ for all $x\in E$ whenever $n\ge N$. By the definition of ideal convergence, for all $x\in E$, we have

*I*is an admissible ideal, the right-hand side of the relation (3) belongs to

*I*, we have

This completes the proof of the theorem. □

**Theorem 13** *The set of all* *I*-*ward continuous functions on a subset* *E* *of* **R** *is a closed subset of the set of all continuous functions on* *E*, *i*.*e*., $\overline{\mathrm{\Delta}iwc(E)}=\mathrm{\Delta}iwc(E)$, *where* $\mathrm{\Delta}iwc(E)$ *is the set of all* *I*-*ward continuous functions on* *E*, $\overline{\mathrm{\Delta}iwc(E)}$ *denotes the set of all cluster points of* $\mathrm{\Delta}iwc(E)$.

*Proof*Let

*f*be an element in $\overline{\mathrm{\Delta}iwc(E)}$. Then there exists a sequence $({f}_{n})$ of points in $\mathrm{\Delta}iwc(E)$ such that ${lim}_{n\to \mathrm{\infty}}{f}_{n}=f$. To show that

*f*is

*I*-ward continuous, consider an

*I*-quasi-sequence $({x}_{n})$ of points in

*E*. Since $({f}_{n})$ converges to

*f*, there exists a positive integer

*N*such that for all $x\in E$ and for all $n\ge N$, $|{f}_{n}(x)-f(x)|<\frac{\epsilon}{3}$. By the definition of an ideal for all $x\in E$, we have

*I*is an admissible ideal, the right-hand side of the relation (3) belongs to

*I*, we have

This completes the proof of the theorem. □

We note that [59], Theorem 6, [56], Theorem 10, and [7], Theorem 2.2, Theorem 3.9, supply some other closed subsets of the set of all continuous functions.

**Corollary 10** *The set of all* *I*-*ward continuous functions on a subset* *E* *of* **R** *is a complete subspace of the space of all continuous functions on* *E*.

*Proof* The proof follows from the preceding theorem. □

In this paper, two new concepts, namely the concept of *I*-ward continuity of a real function and the concept of *I*-ward compactness of a subset of **R**, were introduced and investigated. In this investigation, we have obtained theorems related to *I*-ward continuity, *I*-ward compactness, compactness, sequential continuity, and uniform continuity. We also introduced and studied some other continuities involving *I*-quasi-Cauchy sequences, statistical sequences, and convergent sequences of points in **R**. The present work also contains a generalization of results of the paper [1], and some results in [2] and [4].

## Declarations

### Acknowledgements

The authors would like to thank the referees for a careful reading and several constructive comments that have improved the presentation of the results.

## Authors’ Affiliations

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