A function f is continuous if and only if, for each point x0 in the domain, limn→∞f(xn)=f(x0), whenever limn→∞xn=x0. This is equivalent to the statement that (f(xn)) is a convergent sequence whenever (xn) is convergent. The concept of slowly oscillating continuity is defined in the sense that a function f is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, (f(xn)) is slowly oscillating whenever (xn) is slowly oscillating. A sequence (xn) of points in R is slowly oscillating if limλ→1+lim―nmaxn+1≤k≤[λn]|xk-xn|=0, where [λn] denotes the integer part of λn. Using ɛ>0's and δ's, this is equivalent to the case when, for any given ɛ>0, there exist δ=δ(ɛ)>0 and N=N(ɛ) such that |xm−xn|<ɛ if n≥N(ɛ) and n≤m≤(1+δ)n. A new type compactness is also defined and some new results related to compactness are obtained.

1. Introduction

The notion of statistical convergence was
introduced by Fast [1] and has been investigated in [2]. In [3], Zygmund
called it “almost convergence” and established a relation between it
and strong summability.

Following the idea given in the 1946
American Mathematical Monthly problem [4], a number of authors including Posner
[5], Iwiński [6], Srinivasan [7], Antoni [8], Antoni and Šalát [9], Spigel
and Krupnik [10] have studied A continuity
defined by a regular summability matrix A. Some authors (Öztürk [11], Savaş and Das [12],
Borsík and Šalát [13]) have studied A continuity for
methods of almost convergence or for related methods.

Recently, Connor and Grosse-Erdman [14] have given sequential
definitions of continuity for real functions calling it G continuity
instead of A continuity and
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 cG, into R where s is the set of
all sequences of points in R. A sequence x=(xn) is said to be G convergent to ℓ if x∈cG and G(x)=ℓ. In particular, lim denotes the
limit function limx=limnxn on the linear
space c. A method G is called
regular if every convergent sequence x=(xn) is G convergent with G(x)=limx. A method is called subsequential if whenever x is G convergent with G(x)=ℓ then there is a
subsequence (xnk) of x with limkxnk=ℓ.

The purpose of this paper is to introduce a concept of
slowly oscillating continuity which cannot be given by means of any G as in [14] and
prove that slowly oscillating continuity implies ordinary continuity and so
statistical continuity and is implied by uniform continuity; and introduce
several other types of continuities; and introduce some new types of compactness.

2. Slowly Oscillating Continuity

It is known that a sequence (xn) of points in R is slowly
oscillating if limλ→1+lim¯nmaxn+1≤k≤[λn]|xk−xn|=0 in which [λn] denotes the
integer part of λn. Using
ε>0's and δ's,
this is equivalent to the case when for any given ε>0, there exists δ=δ(ε)>0 and N=N(ε) such that |xm−xn|<ε if n≥N(ε) and n≤m≤(1+δ)n. It is well known that a function f is continuous
if and only if, for each point x0 in the domain, limn→∞f(xn)=f(x0) whenever limn→∞xn=x0. This is equivalent to the statement that (f(xn)) is a convergent
sequence whenever (xn) is convergent.
As a result of completeness, when the domain of the function is all of R, or a complete subset of R, this is also equivalent to the fact
that
(f(xn)) is a Cauchy
sequence whenever (xn) is Cauchy. This
suggests that we might introduce a concept of slowly oscillating continuity as
defined in the sense that a function f is slowly
oscillating continuous if it transforms slowly oscillating sequences to slowly
oscillating sequences, that is, (f(xn)) is slowly
oscillating whenever (xn) is slowly
oscillating.

We note that the sum of two slowly oscillating
continuous functions is slowly oscillating continuous and that the composite of
two slowly oscillating continuous functions is slowly oscillating continuous;
but the product of two slowly oscillating continuous functions needs not be
slowly oscillating continuous as can be seen by considering product of the
slowly oscillating function f(x)=x with itself.

In connection with slowly oscillating sequences and
convergent sequences, the problem arises to investigate the following types of continuity of functions on R where c denotes the set
of all convergent sequences and w denotes the set
of all slowly oscillating sequences of points in R.

Slowly
oscillating continuity of f on R.

(xn)∈w⇒(f(xn))∈c.

(xn)∈c⇒(f(xn))∈c.

limn→∞f(xn)=f(x0) whenever limn→∞xn=x0.

(xn)∈c⇒(f(xn))∈w.

Uniform
continuity of f on R.

It is clear that (c) is equivalent
to (cp1) for each x0∈R. It is easy to see that (wc) implies (d); (wc) implies (w); and (w) implies (d). Now we give a proof of the implication (w) implies (c) in the
following.

Theorem 2.1.

If f is slowly
oscillating continuous on a subset E of
R, then it is continuous on
E in the ordinary
sense.

Proof.

Let (xn) be any
convergent sequence with limk→∞xk=x0. Then the sequence (x1,x0,x2,x0,…,x0,xn,x0,…) also converges
to x0. Then the sequence (x1,x0,x2,x0,…,x0,xn,x0,…) is slowly
oscillating hence, by the hypothesis, the sequence (f(x1),f(x0),f(x2),f(x0),…,f(x0),f(xn),f(x0),…) is slowly
oscillating. It follows from this that (f(x1),f(x0),f(x2),f(x0),…,f(x0),f(xn),f(x0),…) converges to f(x0), so the sequence (f(x1),f(x2),…,f(xn),…) also converges
to f(x0). This completes the proof.

The converse is not always true for the function f(x)=x2 is an example.

Corollary 2.2.

If f is slowly
oscillating continuous, then it is statistically continuous.

Corollary 2.3.

If (xn) is
statistically convergent and slowly oscillating and
f is a slowly
oscillating continuous function, then
(f(xn)) is a convergent
sequence.

Theorem 2.4.

If a function
f on a subset
E of
R is uniformly
continuous, then it is slowly oscillating continuous on
E.

Proof.

Let f be a uniformly
continuous function and let
x=(xn) be any slowly
oscillating sequence of points in E. To prove that f((xn)) is slowly
oscillating, take any ε>0. Uniform continuity of f implies that
there exists a δ>0 such that |f(x)−f(y)|<ε, whenever |x−y|<δ. Since (xn) is slowly
oscillating, for this δ>0, there exist a δ1>0 and N=N(δ)=N1(ε) such that |xm−xn|<δ if n≥N(δ) and n≤m≤(1+δ1)n. Hence |f(xm)−f(xn)|<ε if n≥N(δ) and n≤m≤(1+δ1)n. It follows from this that (f(xn)) is slowly
oscillating.

It is well known that any continuous function on a
compact set is also uniformly continuous. It is also true for a regular
subsequential method G that any
continuous function on a G sequentially
compact set is also uniformly continuous (see [15] for the definition of G compactness).

Theorem 2.5.

If (fn) is a sequence
of slowly oscillating continuous functions defined on a subset
E of
R and (fn) is uniformly
convergent to a function f, then f is slowly
oscillating continuous on E.

Proof.

Let (xn) be a slowly
oscillating sequence and ε>0. Then there exists a positive integer N such that |fn(x)−f(x)|<ε/3 for all x∈E, whenever n≥N. As fN is slowly
oscillating continuous, there exist a δ>0 and a positive
integer N1 such that |fN(xm)−fN(xn)|<ε3 for n≥N1(ε) and n≤m≤(1+δ)n. Now for n≥N1(ε) and n≤m≤(1+δ)n, we have |f(xm)−f(xn)|≤|f(xm)−fN(xm)|+|fN(xm)−fN(xn)|+|fN(xn)−f(xn)|≤ε3+ε3+ε3=ε. This completes the proof of the
theorem.

Now we can give the definition of slowly
oscillating compactness of a subset of R.

Definition 2.6.

A subset F of R is called
slowly oscillating compact if whenever x=(xn) is a sequence
of points in F there is a
slowly oscillating subsequence z=(xnk) of x.

Any compact subset of R is slowly
oscillating compact. Union of two slowly oscillating compact subsets of R is slowly
oscillating compact. We note that any subset of a slowly oscillating compact
set is also slowly oscillating compact, and so intersection of any slowly
oscillating compact subsets of R is slowly
oscillating compact.

Theorem 2.7.

For any regular subsequential method G, if a subset F of R is G sequentially
compact, then it is slowly oscillating compact.

Proof.

The proof can be obtained by
noticing the regularity and subsequentiality of G (see [15] for
the detail of G compactness).

The converse is not necessarily true, for example, {∑k=1n(1/k):n∈N} is slowly
oscillating compact, but it is not G sequentially
compact when G(x):=limx on c.

Theorem 2.8.

Slowly oscillating continuous image of any slowly
oscillating compact subset of R is slowly
oscillating compact.

Proof.

Let f be a slowly
oscillating continuous function on R and
let
F be a slowly
oscillating compact subset of R. Take any sequence y=(yk) of points in f(F). Write yk=f(xk) for each k∈N. As F is assumed to
be slowly oscillating compact, there exists a subsequence z=(zk)=(xnk) of the sequence x=(xk) for which z=(zk) is slowly
oscillating. Since f is slowly
oscillating continuous, the image of the sequence z=(zk), f(z)=(f(zk)) is slowly
oscillating. Since f(z)=(f(zk)) is a
subsequence of the sequence y, the proof is completed.

We add one more compactness defining as saying that a
subset F of R is called
Cauchy compact if whenever x=(xn) is a sequence
of points in F, there is a subsequence z=(zk)=(xnk) of x which is
Cauchy; we see that any Cauchy compact subset of R is also slowly
oscillating compact, and a slowly oscillating continuous image of any Cauchy
compact subset of R is Cauchy
compact.

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