The Approximate Variation of Univariate Uniform Space Valued

Home , Regulated function, Selection principle

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1We recall Helly’s selection principle for monotone functions in Section 4. 2In Section 2, we adopt a more general deﬁnition of regulated functions.

2 dometrics, bounded functions, functions of bounded (Jordan) variation and regulated functions with respect to pseudometrics from the gage, pointwise and uniform convergences of sequences of functions valued in a uniform space. The purpose of this paper is to present a new characterization of a uniform space valued univariate regulated functions and, on the basis of it, establish a pointwise selection principle in terms of the (new notion of) approximate variation for uniform space valued sequences of univariate functions. Further comments on the novelty of our results can be found in Section 2. The paper is organized as follows. In Section 2, we present necessary deﬁnitions and our main results, Theorems 1 and 2. In Section 3, we establish essential properties of the approximate variation (Lemmas 1 and 2), prove Theorem 1, and present illustrating examples including Dirichlet’s function. In Section 4, we prove Theorem 2, show that its main assumption (2.2) is necessary for the uniform convergence, but not necessary for the pointwise convergence of sequences of functions, and comment on the applicability of Theorem 2 to sequences of non-regulated functions.

2. Definitions and main results We begin by reviewing certain definitions and facts needed for our results. Throughout the paper we assume (if not stated otherwise) that the pair (X, U) is a Hausdorff uniform space with the gage of pseudometrics {dp}p∈P (on X) of the uniformity U for some index set P (cf. [26, Chapter 6]). For the reader’s convenience, we describe this assumption in more detail (I)–(IV).

(I) Given p ∈ P, the function dp : X × X → [0, ∞) is a pseudometric on X, i.e., for all x, y, z ∈ X, it satisﬁes the three conditions: dp(x, x) = 0, dp(y, x)= dp(x, y), and dp(x, y) ≤ dp(x, z)+ dp(z,y) (triangle inequality).

(II) The family {dp}p∈P of pseudometrics on X is the gage of the uniformity U for X if {dp}p∈P is the family of all pseudometrics which are uniformly continuous on X × X relative to the product uniformity derived from U. Recall ([26, Theorem 6.11]) that a pseudometric d on X is uniformly continuous on X × X relative to the product uniformity if and only if the set U(d,ε) = {(x, y) ∈ X × X : d(x, y) < ε} belongs to U for all ε > 0. The uniform space (X, U) is said to be Hausdorﬀ if conditions x, y ∈ X and dp(x, y) = 0 for all p ∈ P imply x = y. (III) Every family F of pseudometrics on X generates a uniformity U (by [26, Theorem 6.15]). A direct description of the gage {dp}p∈P of U

3 generated by F is as follows ([26, Theorem 6.18]): the family {U(d,ε): d ∈F and ε> 0} is a subbase for the uniformity U; thus, given a pseudometric d on X, d ∈ {dp}p∈P if and only if, for each ε > 0, there are n δ > 0, n ∈ N and p1,...,pn ∈ P such that i=1 U(dpi , δ) ⊂ U(d,ε). Fur- thermore ([26, Theorem 6.19]), if {dp}p∈P is the gage of U, then the family T {U(dp,ε): p ∈ P and ε> 0} is a base for the uniformity U (i.e., for every U ∈ U there are p ∈ P and ε> 0 such that U(dp,ε) ⊂ U). (IV) “Each concept which is based on the notion of a uniformity can be described in terms of a gage because each uniformity is completely determined by its gage” (cf. [26, Theorem 6.19]). A particular case of a uniform space is a metric space (X,d) with metric d. The standard base of the uniformity U for X is the family {U(d,ε): ε> 0}, where U(d,ε) is as in (II). The uniformity U has a countable base, e.g., the family {U(d, 1/k): k ∈ N}. Two more examples are in order. First, the family {Uα : α ∈ R} with Uα = {(x, y) ∈ R × R : x = y or x>α, y>α} is a base of a uniformity on R. Second, let CB(X; R) be the family of all continuous bounded functions from a completely regular topological space X into R and, given p ∈ P ≡ CB(X; R), deﬁne dp(x, y) = |p(x) − p(y)| for x, y ∈ X. The family {dp : p ∈ P} generates a gage for a uniformity on X which is compatible with the topology on X. For more examples, we refer to [4, 20, 26, 34]. For the sake of brevity, we often write X in place of (X, U). Given a nonempty set T (in what follows, T ⊂ R), we denote by XT the family of all functions f : T → X mapping T into X and equip it with (extended-valued) uniform pseudometrics

T dT,p(f,g) = sup dp(f(t),g(t)), f,g ∈ X , p ∈ P. t∈T

T For p ∈ P, the oscillation of f ∈ X with respect to pseudometric dp is the quantity |f(T )|p = sup dp(f(s), f(t)) ∈ [0, ∞], s,t∈T also known as the dp-diameter of the image f(T )= {f(t): t ∈ T } ⊂ X. We T denote by Bp(T ; X) = {f ∈ X : |f(T )|p < ∞} the family of all dp-bounded functions, and by B(T ; X)= p∈P Bp(T ; X)—the family of all bounded functions mapping T into X. T

4 ∞ Recall that a sequence of points {xj}≡{xj}j=1 ⊂ X converges in the uniform space X to an element x ∈ X (as j →∞) if and only if dp(xj, x) → 0 as j → ∞ for all p ∈ P. Since X is Hausdorﬀ, the limit x is determined uniquely. A subset Y of the uniform space X is said to be sequentially compact (relatively sequentially compact) if each sequence of points from Y has a subsequence which converges in X to an element of Y (to an element of X, respectively). ∞ T Suppose we have a sequence of functions {fj}≡{fj}j=1 ⊂ X and T f ∈ X . If p ∈ P, we write: (a) fj →p f on T if dp(fj(t), f(t)) → 0 as j → ∞ for all t ∈ T , and (b) fj ⇒p f on T if dT,p(fj, f) → 0 as j → ∞. We denote by: (c) fj → f on T the pointwise (or everywhere) convergence of {fj} to f, i.e., fj →p f on T for all p ∈ P, and (d) fj ⇒ f on T the uniform convergence of {fj} to f, i.e., fj ⇒p f on T for all p ∈ P. Clearly, (d) implies (c), but not vice versa. T A sequence of functions {fj} ⊂ X is said to be pointwise relatively sequentially compact on T if the set Y = {fj(t): j ∈ N} is relatively sequentially compact in X for all t ∈ T . From now on, we assume that T is a nonempty subset of the reals R. Given f ∈ XT and p ∈ P, the (Jordan-type) variation of f with respect to pseudometric dp is the quantity (e.g., [6], [10], [31], [32])

Vp(f, T ) = sup dp(f(ti), f(ti−1)) ∈ [0, ∞], π i=1 X where the supremum is taken over all partitions π of T , i.e., m ∈ N and m π = {ti}i=0 ⊂ T such that ti−1 ≤ ti for all i = 1, 2,...,m. We denote by T BVp(T ; X) = {f ∈ X : Vp(f, T ) < ∞} the family of all functions from T into X of bounded dp-variation, and by BV(T ; X) = p∈P BVp(T ; X)—the family of all functions of bounded variation (or BV functions, for short). T The following deﬁnition is crucial for the whole subsequent material.

Deﬁnition 1. The approximate variation of a function f ∈ XT is the two- parameter family {Vε,p(f, T ): ε> 0 and p ∈P} ⊂ [0, ∞] of ε-dp-variations Vε,p(f, T ) of f, given for ε> 0 and p ∈ P, by

Vε,p(f, T ) = inf Vp(g, T ): g ∈ BVp(T ; X) and dT,p(f,g) ≤ ε (2.1) with the convention that inf ∅ = ∞.

5 T The notion of ε-variation Vε(f, T ) of a function f ∈ X with T = [a, b] a closed interval in R and X = RN was introduced by Fraˇnková[21, Defini- tion 3.2]. For any T ⊂ R and a metric space (X,d), this notion was considered and extended by the authors in [14, Sections 4 and 6], and the corresponding approximate variation was thoroughly studied by the first author in [13] (in particular, Example 3.10 from [13] shows that the infimum in (2.1) is in general not attained). The notion of ε-variation was also generalized by the authors for metric space valued bivariate functions in [15]. The notion of the approximate variation is useful in characterizing regulated metric space valued functions in the usual sense ([13], [14], [21]). For our purposes, we adopt the following more general definition. T Given p ∈ P, a function f ∈ X is said to be dp-regulated on T (in symbols, f ∈ Regp(T ; X)) if it satisfies the dp-Cauchy conditions at every left limit point of T and every right limit point of T . More explicitly, given τ ∈ T , which is a left limit point for T (i.e., T ∩ (τ − δ, τ) =6 ∅ for all δ > 0), ′ we have dp(f(s), f(t)) → 0 as T ∋ s, t → τ − 0, and similarly, given τ ∈ T , which is a right limit point for T (i.e., T ∩ (τ ′, τ ′ + δ) =6 ∅ for all δ > 0), ′ T we have dp(f(s), f(t)) → 0 as T ∋ s, t → τ + 0. A function f ∈ X is said to be regulated on T if it is dp-regulated on T for all p ∈ P. We set

Reg(T ; X) = p∈P Regp(T ; X). It is to be noted that a regulated function need not be bounded in general (for instance, the function f ∈ RT , given on T the set T = [0, 1] ∪{2 − (1/n): n =2, 3,... } by f(t)= t for 0 ≤ t ≤ 1 and f(2 − (1/n)) = n for n =2, 3,... , is regulated on T in the above sense, but not bounded). In the case when T = I = [a, b] is a closed interval in R and X is complete we have: f ∈ Reg(I; X) if and only if the (left) limit of f(t) as t → τ − 0 exists in X at each point τ ∈ (a, b] and the (right) limit of f(t) as t → τ ′ + 0 exists in X at each point τ ′ ∈ [a, b) (cf. [10, Section 4]). Our ﬁrst result is a characterization of regulated functions in terms of the approximate variation (containing similar results from [14, equality (4.2)] and [21, Proposition 3.4] as particular cases): Theorem 1. Suppose ∅ =6 T ⊂ R and (X, U) is a Hausdorﬀ uniform space. T Given p ∈ P, {f ∈ X : Vε,p(f, T ) < ∞ ∀ ε> 0} ⊂ Regp(T ; X), and this inclusion turns into the equality for T = I = [a, b]. Consequently, I Reg(I; X)= {f ∈X : Vε,p(f,I)<∞ for all ε>0 and p ∈ P}. As it was already mentioned in the Introduction, there is a relationship between ‘characterizations of regulated functions’ and ‘pointwise selection

6 principles’. The second main result is a pointwise selection principle for uniform space valued functions in terms of the approximate variation:

Theorem 2. Suppose ∅ =6 T ⊂ R and (X, U) is a Hausdorﬀ uniform space with an at most countable gage of pseudometrics {dp}p∈P of the uniformity U. T If {fj} ⊂ X is a pointwise relatively sequentially compact sequence of functions on T such that

lim sup Vε,p(fj, T ) < ∞ for all ε> 0 and p ∈ P, (2.2) j→∞ then there is a subsequence of {fj} which converges pointwise on T to a function f ∈ B(T ; X) ∩ Reg(T ; X).

The novelty of this theorem is threefold. First, functions fj take their values in a uniform space, which (as it was seen earlier) is more general than a metric space. Second, condition (2.2) is necessary for uniformly convergent sequences {fj} (but not for pointwise convergent sequences {fj}). Third, Theorem 2 may be applied to sequences {fj} of non-regulated functions. These issues are addressed in Section 4.

3. Properties of the approximate variation 3.1. Bounded functions and functions of bounded variation. T Given f,g ∈ X , s, t ∈ T , and p ∈ P, by the triangle inequality for dp, we ﬁnd dT,p(f,g) ≤|f(T )|p + dp(f(t),g(t))+ |g(T )|p (3.1) and dp(f(s), f(t)) ≤ dp(g(s),g(t))+2dT,p(f,g) (3.2) and so, the deﬁnition of the dp-oscillation and (3.2) imply

|f(T )|p ≤|g(T )|p +2dT,p(f,g). (3.3)

By (3.1) and (3.3), we have dT,p(f,g) < ∞ for all f,g ∈ Bp(T ; X) with p ∈ P T T and, given a constant function c ∈ X , Bp(T ; X)= {f ∈ X : dT,p(f,c) < ∞} T for all p ∈ P, and so, B(T ; X)= {f ∈ X : dT,p(f,c) < ∞ for all p ∈ P}.

7 Clearly,

T |f(T )|p ≤ Vp(f, T ) for all f ∈ X and p ∈ P, (3.4) and so,

BVp(T ; X) ⊂ Bp(T ; X) and BV(T ; X) ⊂ B(T ; X).

T Given f ∈ X and p ∈ P, the functional Vp(·, ·) has the following two properties: (i) additivity of Vp(f, · ) in the second variable (cf. [6], [12]):

Vp(f, T )= Vp(f, T ∩ (−∞, t]) + Vp(f, T ∩ [t, ∞)) for all t ∈ T ; (3.5)

(ii) sequential lower semicontinuity of Vp( · , T ) in the ﬁrst variable (cf. [10]):

T if {fj} ⊂ X and fj →p f on T , then Vp(f, T ) ≤ lim inf Vp(fj, T ). (3.6) j→∞

3.2. The approximate variation. The essential properties of the approximate variation are gathered in

Lemma 1. Given f ∈ XT and p ∈ P, we have:

(a) the function ε 7→ Vε,p(f, T ):(0, ∞) → [0, ∞] is nonincreasing, and so, 3 Vε+0,p(f, T ) ≤ Vε,p(f, T ) ≤ Vε−0,p(f, T ) for all ε> 0;

(b) if ∅ =6 T1 ⊂ T2 ⊂ T and ε> 0, then Vε,p(f, T1) ≤ Vε,p(f, T2);

(d) |f(T )|p ≤ Vε,p(f, T )+2ε for all ε> 0; − + (e) if ε> 0, t ∈ T , Tt = T ∩ (−∞, t] and Tt = T ∩ [t, ∞), then

− + − + Vε,p(f, Tt )+ Vε,p(f, Tt ) ≤ Vε,p(f, T ) ≤ Vε,p(f, Tt )+ Vε,p(f, Tt )+2ε.

T Proof. (a) Suppose 0 < ε1 < ε2. If g ∈ X and dT,p(f,g) ≤ ε1, then dT,p(f,g) ≤ ε2, and so, deﬁnition (2.1) implies Vε2,p(f, T ) ≤ Vε1,p(f, T ). T (b) Since T1 ⊂ T2, we have dT1,p(f,g) ≤ dT2,p(f,g) for all g ∈ X , whence, by deﬁnition (2.1), Vε,p(f, T1) ≤ Vε,p(f, T2).

3 As usual, Vε+0, p(f,T ) and Vε−0, p(f,T ) are the limits from the right and from the left of the function ε 7→ Vε, p(f,T ), respectively.

8 (c) By (a), the quantity Cp = limε→+0 Vε,p(f, T ), which actually is equal to supε>0 Vε,p(f, T ), is well-defined in [0, ∞]. Assume first that f ∈ BVp(T ; X). By (2.1), Vε,p(f, T ) ≤ Vp(f, T ) for all ε> 0, and so, Cp ≤ Vp(f, T ) < ∞. To prove the reverse inequality, we apply the definition of Cp: for every η > 0 there is δ = δ(η) > 0 such that Vε,p(f, T ) ≤ Cp + η for all ε ∈ (0, δ). If ∞ {εk}k=1 is a sequence in (0, δ) such that εk → 0 as k → ∞, then, for every k ∈ N, the definition of Vεk,p(f, T ) yields the existence of gk ∈ BVp(T ; X) such that dT,p(f,gk) ≤ εk and Vp(gk, T ) ≤ Cp + η. Since εk → 0, gk ⇒p f on T , hence gk →p f on T , and so, by (3.6),

Vp(f, T ) ≤ lim inf Vp(gk, T ) ≤ Cp + η. k→∞

By the arbitrariness of η > 0, Vp(f, T ) ≤ Cp. Thus, Cp and Vp(f, T ) are ﬁnite or not ﬁnite simultaneously, and Cp = Vp(f, T ), which proves (c).

(d) We may assume that ε> 0 is such that Vε,p(f, T ) is ﬁnite. By deﬁni- tion (2.1), given η > 0, there is g = gη ∈ BVp(T ; X) such that dT,p(f,g) ≤ ε and Vp(g, T ) ≤ Vε,p(f, T )+ η. Now, (3.3) and (3.4) imply

|f(T )|p ≤|g(T )|p +2dT,p(f,g) ≤ Vp(g, T )+2ε ≤ Vε,p(f, T )+ η +2ε, and the inequality in (d) follows due to the arbitrariness of η > 0. (e) First, we prove the left-hand side inequality, in which we may assume that Vε,p(f, T ) is ﬁnite. By deﬁnition (2.1), for every η > 0 there is g = gη ∈ BVp(T ; X) such that dT,p(f,g) ≤ ε and Vp(g, T ) ≤ Vε,p(f, T )+ η. We − − + + set g (s) = g(s) for s ∈ Tt and g (s) = g(s) for s ∈ Tt , and note that − + − − + + g (t)= g(t)= g (t). Since g ∈ BVp(Tt ; X), g ∈ BVp(Tt ; X),

− + d − (f,g ) ≤ dT,p(f,g) ≤ ε and d + (f,g ) ≤ dT,p(f,g) ≤ ε, Tt ,p Tt ,p

− − − + + + deﬁnition (2.1) implies Vε,p(f, Tt ) ≤ Vp(g , Tt ) and Vε,p(f, Tt ) ≤ Vp(g , Tt ), and so, by the additivity property (3.5), we ﬁnd

− + − − + + − + Vε,p(f, Tt )+ Vε,p(f, Tt ) ≤ Vp(g , Tt )+ Vp(g , Tt )= Vp(g, Tt )+ Vp(g, Tt )

= Vp(g, T ) ≤ Vε,p(f, T )+ η for all η > 0.

This establishes the left-hand side inequality. In order to prove the right-hand side inequality, we may assume that − + − + Vε,p(f, Tt ) and Vε,p(f, Tt ) are ﬁnite. Note that if Tt ={t}, then Tt =T , and

9 + − if Tt ={t}, then Tt =T ; in both these cases the inequality is clear. Assume − + that Tt =6 {t} and Tt =6 {t}, so that T ∩ (−∞, t) =6 ∅ and T ∩ (t, ∞) =6 ∅. − + − − By definition (2.1), for every η > 0 and η > 0 there are g ∈ BVp(Tt ; X) + + − + and g ∈ BVp(T ; X) with the properties: d − (f,g )≤ε, d + (f,g )≤ε, t Tt ,p Tt ,p − − − − + + + + Vp(g , Tt ) ≤ Vε,p(f, Tt )+ η and Vp(g , Tt ) ≤ Vε,p(f, Tt )+ η . Given x ∈ X (to be specified below), we define g ∈ BVp(T ; X) by

g−(s) if s ∈ T ∩ (−∞, t), g(s)= x if s = t,  + g (s) if s ∈ T ∩ (t, ∞). Suppose that the following two inequalities are already established:

− − − − Vp(g, Tt ) ≤ Vp(g , Tt )+ dp(g(t),g (t)) (3.7) and + + + + Vp(g, Tt ) ≤ Vp(g , Tt )+ dp(g(t),g (t)). (3.8)

By the additivity property (3.5) of Vp(·, ·), these inequalities imply

− + Vp(g, T )= Vp(g, Tt )+ Vp(g, Tt ) − − − ≤ Vε,p(f, Tt )+ η + dp(x, g (t)) + + + + Vε,p(f, Tt )+ η + dp(x, g (t)). (3.9)

Now, we put x = g−(t) (by symmetry, we may have put x = g+(t) as well). − − + + Since g = g on Tt and g = g on T ∩ (t, ∞) ⊂ Tt , we get

− + dT,p(f,g) ≤ max d − (f,g ),d + (f,g ) ≤ ε. (3.10) Tt ,p Tt ,p Noting that (cf. (3.9))

+ − + − + dp(x, g (t)) = dp(g (t),g (t)) ≤ dp(g (t), f(t)) + dp(f(t),g (t)) − + ≤ d − (g , f)+ d + (f,g ) ≤ ε + ε =2ε, Tt ,p Tt ,p we conclude from (2.1), (3.9) and (3.10) that

− − + + Vε,p(f, T ) ≤ Vp(g, T ) ≤ Vε,p(f, Tt )+ η + Vε,p(f, Tt )+ η +2ε, and the inequality in (e) follows due to the arbitrariness of η− >0 and η+ >0.

10 m − It remains to establish (3.7) (since (3.8) is similar). Let {ti}i=0 ⊂ Tt be − − a partition of Tt , i.e., t0 ≤ t1 ≤ ... ≤ tm−1 < tm = t. Since g(s)= g (s) for all s ∈ T with s < t, we have

m m−1

dp(g(ti),g(ti−1)) = dp(g(ti),g(ti−1)) + dp(g(tm),g(tm−1)) i=1 i=1 X mX−1 − − − − = dp(g (ti),g (ti−1)) + dp(g (tm),g (tm−1)) i=1 X − − + dp(g(tm),g(tm−1))−dp(g (tm),g (tm−1))

− − − − − ≤ Vp(g , Tt )+ dp(g(t),g (tm−1))−dp(g (t),g (tm−1)) ≤ V (g−, T −)+ d (g(t),g−(t)), p t p where the last inequality is due to the triangle inequality for dp. Taking the − supremum over all partitions of Tt , we obtain inequality (3.7).

Remark 1. Inequalities in Lemma 1 (e) are sharp (cf. [13, Example 3.2]).

3.3. Regulated functions and Dirichlet’s function. Now, we are in a position to prove Theorem 1.

T Proof of Theorem 1. Suppose p ∈ P, and f ∈ X is such that Vε,p(f, T ) is finite for all ε > 0. Given τ ∈ T , which is a left limit point for T , let us ′ show that dp(f(s), f(t)) → 0 as T ∋ s, t → τ − 0 (the arguments for τ ∈ T being a right limit point for T are similar). For any ε> 0, we define the ε-dp- − − variation function by ϕε,p(t)= Vε,p(f, Tt ), t ∈ T , where Tt = T ∩ (−∞, t]. − − By Lemma 1 (b), given s, t ∈ T with s ≤ t (and so, Ts ⊂ Tt ), we find

0 ≤ ϕε,p(s) ≤ ϕε,p(t) ≤ Vε,p(f, T ) < ∞, i.e., ϕε,p : T → [0, ∞) is a bounded and nondecreasing function. Conse- quently, the left limit

lim ϕε,p(t) = sup ϕε,p(t) exists in [0, ∞). T ∋t→τ−0 t∈T ∩(−∞,τ)

It follows that there is δ = δ(ε) > 0 such that |ϕε,p(t) − ϕε,p(s)| < ε for all s, t ∈ T ∩ (τ − δ, τ). Now, assume that s, t ∈ T ∩ (τ − δ, τ) are arbitrary such

11 − − − − that s < t. Lemma 1 (e) (in which T is replaced by Tt , so that (Tt )s = Ts − + − and (Tt )s = T ∩ [s, t] for s ∈ Tt ) implies

− − Vε,p(f, T ∩ [s, t]) ≤ Vε,p(f, Tt ) − Vε,p(f, Ts )= ϕε,p(t) − ϕε,p(s) < ε.

By the deﬁnition of Vε,p(f, T ∩ [s, t]), there is g = gε,p ∈ BVp(T ∩ [s, t]; X) such that dT ∩[s,t],p(f,g) ≤ ε and Vp(g, T ∩ [s, t]) ≤ ε, and so, (3.2) yields dp(f(s), f(t)) ≤ dp(g(s),g(t))+2dT ∩[s,t],p(f,g) ≤ Vp(g, T ∩ [s, t]) + 2ε ≤ 3ε.

This completes the proof of the equality limT ∋s,t→τ−0 dp(f(s), f(t)) = 0.

Suppose now that f ∈ Regp(I; X) with I = [a, b]. By virtue of [10, 4 I Lemma 4], there is a sequence of step functions {gj} ⊂ X such that gj ⇒p f on I, and so, given ε> 0, there is j(ε) ∈ N such that dI,p(f,gj(ε)) ≤ ε. Since gj(ε) ∈ BVp(I; X), deﬁnition (2.1) yields Vε,p(f,I) ≤ Vp(gj(ε),I) < ∞, which was to be proved.

As an illustration of Theorem 1, we present an example.

Example 1. Let I = [a, b], x, y ∈ X, x =6 y, and Q be the set of all rational I numbers. We denote by f = Dx,y ∈ X the Dirichlet function given by

Dx,y(t)= x if t ∈ I ∩ Q, and Dx,y(t)= y if t ∈ I \ Q. (3.11)

It is clearly seen that f∈ / Reg(I; M): in fact, since X is Hausdorﬀ and x =6 y, there is p0 ∈ P such that dp0 (x, y) > 0, and so, if, say, a < τ ≤ b, then for all δ ∈ (0, τ − a), s ∈ (τ − δ, τ) ∩ Q and t ∈ (τ − δ, τ) \ Q, we have dp0 (f(s), f(t)) = dp0 (x, y) > 0, so that the limit of dp0 (f(s), f(t)) as I ∋ s, t → τ − 0 does not exist.

We claim that, for ε> 0 and all p ∈ P with dp(x, y) > 0, we have

∞ if ε

(the value Vε,p(f,I) for dp(x, y)/2 ≤ ε

4 I Recall that g ∈ X is a step function if, for some partition a = t0 < t1 < t2 <...< tm−1 < tm = b of I = [a,b], g takes a constant value on each open interval (ti−1,ti), i =1, 2,...,m. Clearly, g ∈ BVp(I; X) for all p ∈P.

12 we may set g(t)= x (or g(t)= y) for all t ∈ I, so that dI,p(f,g)= dp(x, y) ≤ ε and, by (2.1), 0 ≤ Vε,p(f,I) ≤ Vp(g,I) = 0. Suppose 0 <ε

dp(x, y)= dp(f(t2i), f(t2i−1)) ≤ dp(g(t2i),g(t2i−1))+2ε, whence, by the deﬁnition of Vp(g,I), m

Vp(g,I) ≥ dp(g(t2i),g(t2i−1)) ≥ (dp(x, y) − 2ε)m ∀ m ∈ N. i=1 X 3.4. Uniform convergence vs. pointwise convergence. The next lemma shows the interplay of the approximate variation and the uniform convergence. Lemma 2. Let ∅ =6 T ⊂ R and (X, U) be a Hausdorﬀ uniform space. T T Suppose p ∈ P, f ∈ X , {fj} ⊂ X , and fj ⇒p f on T . We have:

(a) Vε+0,p(f, T )≤lim inf Vε,p(fj, T )≤lim sup Vε,p(fj, T )≤Vε−0,p(f, T ), ε>0; j→∞ j→∞

(b) if Vε,p(fj, T )<∞ for all ε>0 and j ∈N, then Vε,p(f, T )<∞ for all ε>0. Proof. (a) Only the first and the last inequalities are to be established. 1. In order to prove the first inequality, we may assume (passing to a suitable subsequence of {fj} if necessary) that its right-hand side is equal to Cp = limj→∞ Vε,p(fj, T ) < ∞. Let η > 0 be given arbitrarily. Then, there is j0 = j0(η) ∈ N such that Vε,p(fj, T ) ≤ Cp + η for all j ≥ j0. By the definition of Vε,p(fj, T ), for every j ≥ j0 there is gj ∈ BVp(T ; X) (also depending on η and p) such that dT,p(fj,gj)≤ε and Vp(gj, T )≤Vε,p(fj, T )+η. Since fj ⇒p f on T , we find dT,p(fj, f) → 0 as j →∞, and so, there is j1 = j1(η) ∈ N such that dT,p(fj, f) ≤ η for all j ≥ j1. Taking into account that

dT,p(f,gj) ≤ dT,p(f, fj)+ dT,p(fj,gj) ≤ η + ε for all j ≥ max{j0, j1}, and applying deﬁnition (2.1), we get

Vη+ε,p(f, T ) ≤ Vp(gj, T ) ≤ Vε,p(fj, T )+ η ≤ (Cp + η)+ η = Cp +2η.

Passing to the limit as η → +0, we arrive at Vε+0,p(f, T ) ≤ Cp, which was to be proved.

13 2. To establish the third inequality in (a), we may assume (with no loss of generality) that Vε−0,p(f, T ) is ﬁnite. Given η > 0, there is δ = δ(η,ε) ∈ (0,ε) ′ such that Vε′,p(f, T ) ≤ Vε−0,p(f, T )+η for all ε ∈ (ε−δ, ε). Since fj ⇒p f on ′ ′ T , for every ε ∈ (ε − δ, ε) there is j0 = j0(ε ,ε) ∈ N such that dT,p(fj, f) ≤ ′ ε − ε for all j ≥ j0. By the deﬁnition of Vε′,p(f, T ), given j ∈ N, there is ′ ′ gj ∈ BVp(T ; X) (also depending on ε and p) such that dT,p(f,gj) ≤ ε and

Vε′,p(f, T ) ≤ Vp(gj, T ) ≤ Vε′,p(f, T )+(1/j).

Hence, limj→∞ Vp(gj, T )= Vε′,p(f, T ). Noting that, for all j ≥ j0,

′ ′ dT,p(fj,gj) ≤ dT,p(fj, f)+ dT,p(f,gj) ≤ (ε − ε )+ ε = ε, we ﬁnd from (2.1) that Vε,p(fj, T ) ≤ Vp(gj, T ) for all j ≥ j0. Thus,

lim sup Vε,p(fj, T ) ≤ lim Vp(gj, T )= Vε′,p(f, T ) ≤ Vε−0,p(f, T )+ η. j→∞ j→∞

It remains to take into account the arbitrariness of η > 0. ′ (b) Let ε > 0 and 0 < ε < ε. Given j ∈ N, since Vε′,p(fj, T ) < ∞, deﬁ- ′ nition (2.1) implies the existence of gj ∈ BVp(T ; X) such that dT,p(fj,gj) ≤ ε ′ and Vp(gj,T, ) ≤ Vε′,p(fj, T )+1. Since fj ⇒p f on T , there is j0 = j0(ε,ε ) ∈ N ′ such that dT,p(fj0 , f) ≤ ε − ε . Now, it follows from

′ ′ dT,p(f,gj0 ) ≤ dT,p(f, fj0 )+ dT,p(fj0 ,gj0 ) ≤ (ε − ε )+ ε = ε

′ and deﬁnition (2.1) that Vε,p(f, T ) ≤ Vp(gj0 , T ) ≤ Vε ,p(fj0 , T )+1 < ∞.

A few comments and examples concerning Lemma 2 are in order.

Example 2. The left limit Vε−0,p(f, T ) in Lemma 2 (a) cannot in general be replaced by Vε,p(f, T ). We demonstrate this in the case of a normed linear space (X, k·k) equipped with the canonical metric d(x, y)= kx − yk, x, y ∈ X (here we omit the subscript p in the notations dT = dT,p, V = Vp and Vε = Vε,p). Let T = I = [a, b] and {xj}, {yj} ⊂ X be two sequences such that kxj − xk→ 0 and kyj − yk→ 0 as j →∞, where x, y ∈ X, x =6 y.

If fj = Dxj ,yj , j ∈ N, and f = Dx,y are Dirichlet functions (3.11) on I, then fj ⇒ f on I, because

dI (fj, f)=sup kfj(t)−f(t)k=max kxj −xk, kyj −yk → 0 as j →∞. t∈I 14 According to (3.12), the values Vε(f,I) are given for ε> 0 by

Vε(f,I)= ∞ if ε< kx−yk/2, and Vε(f,I) = 0 if ε ≥kx−yk/2 (note that if kx−yk/2 ≤ ε< kx−yk, we may set g(t)=(x+y)/2 for all t ∈ I, which implies dI(f,g)= kx − yk/2 ≤ ε, and so, 0 ≤ Vε(f,I) ≤ V (g,I) = 0). Similarly, for (large) j ∈ N,

Vε(fj,I)= ∞ if ε< kxj −yjk/2, and Vε(fj,I) = 0 if ε ≥kxj −yjk/2. Setting ε = kx − yk/2, we ﬁnd

Vε+0(f,I)= Vε(f,I)=0 < ∞ = Vε−0(f,I), whereas, if xj = αjx and yj = αjy with αj = 1+(1/j), j ∈ N, we get ε<αjkx − yk/2 = kxj − yjk/2, which implies Vε(fj,I) = ∞ for all j ∈ N, and so, limj→∞ Vε(fj,I)= ∞. Example 3. In this example, we show that the right-hand side inequality T T in Lemma 2 (a) does not hold in general if {fj} ⊂ X converges to f ∈ X only pointwise on T . Suppose p0 ∈ P is such that Cp0 = infj∈N |fj(T )|p0 > 0 N and f = c (is a constant function) on T . Given 0 <ε

Vε,p0 (fj, T ) ≥|fj(T )|p0 − 2ε ≥ Cp0 − 2ε> 0= Vε,p0 (c, T )= Vε−0,p0 (f, T ).

More speciﬁcally, let x, y ∈ X, x =6 y, p0 ∈ P be such that dp0 (x, y) > 0, ∞ and {τj}j=1 ⊂ (a, b) ⊂ T = [a, b] be such that τj → a as j → ∞. Deﬁning T fj ∈ X by fj(τj)= x and fj(t)= y if t ∈ T \{τj}, we get Cp0 = dp0 (x, y) > 0 and fj →p c ≡ y on T for all p ∈ P. Note that the arguments above are not valid for the uniform convergence: in fact, if p ∈ P and fj ⇒p f = c on T , then, by (3.3),

|fj(T )|p ≤ 2dT,p(fj,c) → 0 as j →∞, and so, Cp ≡ infj∈N |fj(T )|p = 0.

Example 4. Lemma 2 (b) is wrong for the pointwise convergence of {fj} to f. To see this, let T = I = [a, b], x, y ∈ X, x =6 y, and, given j ∈ N, deﬁne I fj ∈ X at t ∈ I by: fj(t) = x if j! · t is integer, and fj(t) = y otherwise. Each fj is a step function on I, so fj ∈ Reg(I; X) and, hence, by Theorem 1, Vε,p(fj,I) < ∞ for all ε > 0 and p ∈ P. Clearly, fj converges (only) pointwise on I to the Dirichlet function f = Dx,y from (3.11), and so, by (3.12), Vε,p(f,I)= ∞ for all p ∈ P with dp(x, y) > 0 and 0 <ε

15 4. Proof of the main result

We denote by Mon(T ; R+) the family of all bounded nondecreasing functions mapping T into R+ = [0, ∞) (where R+ may be replaced by R). It is worthwhile to recall the classical Helly selection principle for an arbitrary set T ⊂ R (cf. [7, Proof of Theorem 1.3]): a uniformly bounded sequence of functions from Mon(T ; R) contains a subsequence which converges pointwise on T to a function from Mon(T ; R).

Proof of Theorem 2. By Lemma 1 (b), given ε> 0, p ∈ P and j ∈ N, the − ε-dp-variation function t 7→ Vε,p(fj, Tt ) is nondecreasing on T (recall that − Tt = T ∩(−∞, t] for t ∈ T ). By assumption (2.2), for every ε> 0 and p ∈ P ∗ there are j (ε,p) ∈ N and C(ε,p) > 0 such that Vε,p(fj, T ) ≤ C(ε,p) for all natural j ≥ j∗(ε,p). Again by Lemma 1 (b),

− ∗ Vε,p(fj, Tt ) ≤ Vε,p(fj, T ) ≤ C(ε,p) for all t ∈ T and j ≥ j (ε,p),

− ∞ and so, the sequence of nondecreasing functions {t 7→ Vε,p(fj, Tt )}j=j∗(ε,p) is uniformly bounded in t ∈ T by constant C(ε,p). We divide the rest of the proof into four steps. With no loss of generality we assume that P = N. 1. Making use of the Cantor diagonal process, let us show that for each ∞ decreasing sequence {εk}k=1 of positive numbers εk → 0 (as k → ∞) there is a subsequence of {fj}, again denoted by {fj}, and for every p ∈ P there ∞ + is a sequence of functions {ϕk,p}k=1 ⊂ Mon(T ; R ) such that

− lim Vεk,p(fj, Tt )= ϕk,p(t) for all t ∈ T and k ∈ N. (4.1) j→∞

− ∞ ∗ We begin with p = 1. The sequence {t 7→ Vε1, 1(fj, Tt )}j=j (ε1,1) from + Mon(T ; R ) is uniformly bounded in t ∈ T by C(ε1, 1), and so, by the Helly ∞ ∞ ∗ selection principle, there are a subsequence {J1,1(j)}j=1 of {j}j=j (ε1,1) and a + function ϕ1,1 ∈ Mon(T ; R ) such that

− lim Vε1, 1 fJ1,1(j), Tt = ϕ1,1(t) for all t ∈ T . j→∞

∗ Now, choose the least number j1,1 ∈ N such that J1,1(j1,1) ≥ j (ε2, 1).

16 ∞ Inductively, assume that k ≥ 2 and a subsequence {Jk−1,1(j)}j=1 of ∞ ∗ ∗ N {j}j=j (ε1,1) and the number jk−1,1 ∈ such that Jk−1,1(jk−1,1) ≥ j (εk, 1) − ∞ have already been constructed. Since {t 7→ Vεk, 1(fJk−1,1(j), Tt )}j=jk−1,1 is a + sequence from Mon(T ; R ), uniformly bounded in t ∈ T by C(εk, 1), the ∞ Helly selection principle implies the existence of a subsequence {Jk,1(j)}j=1 ∞ R+ of {Jk−1,1(j)}j=jk−1,1 and a function ϕk,1 ∈ Mon(T ; ) such that

− lim Vεk, 1 fJk,1(j), Tt = ϕk,1(t) for all t ∈ T . j→∞

∞ ∞ Given k ∈ N, {Jj,1(j)}j=k is a subsequence of {Jk,1(j)}j=1, and so, the di- (1) ∞ ∞ ∞ agonal subsequence {fj }j=1 ≡{fJj,1(j)}j=1 of the original sequence {fj}j=1 satisﬁes the condition

(1) − lim Vεk, 1 fj , Tt = ϕk,1(t) for all t ∈ T and k ∈ N. (4.2) j→∞ Taking into account (4.2), again inductively, assume that p ∈ P, p ≥ 2, (p−1) ∞ (1) ∞ and a subsequence {fj }j=1 of {fj }j=1 (and, hence, of {fj}) satisfying

(p−1) − lim Vεk,p−1 fj , Tt = ϕk,p−1(t) for all t ∈ T and k ∈ N j→∞

(p−1) − ∞ ∗ is already chosen. The sequence {t 7→ Vε1,p(fj , Tt )}j=j (ε1,p) of nondecreasing functions on T is uniformly bounded in t ∈ T by C(ε1,p), and so, by the Helly selection principle, there are a subsequence {f (p−1) }∞ of J1,p(j) j=1 (p−1) ∞ ∗ N N {fj }j=j (ε1,p) (where J1,p : → is a strictly increasing subsequence of ∞ + ∗ R {j}j=j (ε1,p)) and a function ϕ1,p ∈ Mon(T ; ) such that

(p−1) − lim Vε1,p f , Tt = ϕ1,p(t) for all t ∈ T . j→∞ J1,p(j)

∗ Pick the least number j1,p ∈ N such that J1,p(j1,p) ≥ j (ε2,p). Inductively, as- ∞ ∞ ∗ sume now that k ≥ 2 and a subsequence {Jk−1,p(j)}j=1 of {j}j=j (ε1,p) and the ∗ (least) number jk−1,p ∈ N such that Jk−1,p(jk−1,p) ≥ j (εk,p) are already chosen. Since {t 7→ V (f (p−1) , T −)}∞ is a sequence from Mon(T ; R+), εk,p Jk−1,p(j) t j=jk−1,p uniformly bounded in t ∈ T by C(εk,p), the Helly selection principle im- ∞ ∞ plies the existence of a subsequence {Jk,p(j)}j=1 of {Jk−1,p(j)}j=jk−1,p and a + function ϕk,p ∈ Mon(T ; R ) such that

(p−1) − lim Vεk,p f , Tt = ϕk,p(t) for all t ∈ T . j→∞ Jk,p(j) 17 ∞ ∞ Given k ∈ N, {Jj,p(j)}j=k is a subsequence of {Jk,p(j)}j=1, and so, the diagonal subsequence {f (p)}∞ ≡{f (p−1) }∞ of {f (p−1) }∞ satisﬁes the equality j j=1 Jj,p(j) j=1 Jk,p(j) j=1

(p) − lim Vεk,p fj , Tt = ϕk,p(t) for all t ∈ T and k ∈ N. j→∞ (j) ∞ ∞ Finally, the diagonal sequence {fj }j=1, again denoted by {fj}≡{fj}j=1, satisﬁes (4.1). 2. Let Q be an at most countable dense subset of T . Note that any point t ∈ T , which is not a limit point for T (i.e., T ∩ (t − δ, t + δ)= {t} for some + δ > 0), belongs to Q. Since, for any k ∈ N and p ∈ P, ϕk,p ∈ Mon(T ; R ), the set Qk,p ⊂ T of points of discontinuity of ϕk,p is at most countable. Setting

S = Q∪ k∈N p∈P Qk,p, we ﬁnd that S is an at most countable dense subset of T . Furthermore, if S =6 T , then every point t ∈ T \ S is a limit point for T and S S ϕk,p is continuous on T \ S for all k ∈ N and p ∈ P. (4.3)

Since S is at most countable and {fj(s): j ∈ N} is a relatively sequentially compact subset of X for all s ∈ S, with no loss of generality we may assume (applying the Cantor diagonal process and passing to a subsequence of {fj} if necessary) that, for each s ∈ S, fj(s) converges in X as j →∞ to a unique point denoted by f(s) ∈ X, so that f : S → X and limj→∞ dp(fj(s), f(s))=0 for all p ∈ P. If S = T , we turn to step 4 below which completes the proof. 3. Assume that S =6 T and t ∈ T \ S is arbitrary. Let us prove that ∞ {fj(t)}j=1 is a Cauchy sequence in X, i.e.,

lim dp(fj(t), fj′ (t)) = 0 for all p ∈ P. j,j′→∞

Let us ﬁx p ∈ P. Since εk → 0 as k → ∞, given η > 0, choose and ﬁx k = k(η) ∈ N such that εk ≤ η. By (4.3), ϕk,p is continuous at t, and so, by the density of S in T , there is s = s(η,k,t,p) ∈ S such that

|ϕk,p(t) − ϕk,p(s)| ≤ η.

Property (4.1) implies the existence of j1 = j1(η,k,t,s,p) ∈ N such that, for all j ≥ j1, we have

− − |Vεk,p(fj, Tt ) − ϕk,p(t)| ≤ η and |Vεk,p(fj, Ts ) − ϕk,p(s)| ≤ η. (4.4)

18 Supposing (with no loss of generality) that s < t and applying Lemma 1 (e) − − − − − + (in which T is replaced by Tt , so that (Tt )s = Ts and (Tt )s = T ∩ [s, t]), we get − − Vεk,p(fj, T ∩ [s, t]) ≤ Vεk,p(fj, Tt ) − Vεk,p(fj, Ts ) − ≤|Vεk,p(fj, Tt ) − ϕk,p(t)| + |ϕk,p(t) − ϕk,p(s)| − + |ϕk,p(s) − Vεk,p(fj, Ts )| ≤ η + η + η =3η for all j ≥ j1. 1 By the deﬁnition of Vεk,p(fj, T ∩ [s, t]), for each j ≥ j there is a function gj ∈ BVp(T ∩ [s, t]; X), also depending on η, k, t, s and p, such that

dT ∩[s,t],p(fj,gj) ≤ εk and Vp(gj, T ∩ [s, t]) ≤ Vεk,p(fj, T ∩ [s, t]) + η. 1 These inequalities, (3.2) and the deﬁnition of Vp(·, ·) imply, for all j ≥ j ,

dp(fj(s), fj(t)) ≤ dp(gj(s),gj(t))+2dT ∩[s,t],p(fj,gj)

≤ Vp(gj, T ∩ [s, t]) + 2εk ≤ (3η + η)+2η =6η. (4.5) ∞ Being convergent in the uniform space X, the sequence {fj(s)}j=1 is Cauchy (cf. [26, Theorem 6.21]), and so, there is j2 = j2(η,s,p) ∈ N such that ′ 2 3 1 2 dp(fj(s), fj′ (s)) ≤ η for all j, j ≥ j . The number j = max{j , j } depends only on η, t and p, and we ﬁnd, by the triangle inequality for dp,

dp(fj(t), fj′ (t)) ≤ dp(fj(t), fj(s)) + dp(fj(s), fj′ (s)) + dp(fj′ (s), fj′ (t)) ≤ 6η + η +6η = 13η for all j, j′ ≥ j3. ∞ Due to the arbitrariness of p ∈ P, this proves that {fj(t)}j=1 is a Cauchy sequence in X. Taking into account that the set {fj(t): j ∈ N} is relatively ∞ sequentially compact in X, we conclude that the sequence {fj(t)}j=1 has a limit point in X, which we denote by f(t) ∈ X. By [26, Theorem 6.21], a Cauchy sequence in a uniform space converges to its limit point, and so, limj→∞ dp(fj(t), f(t)) = 0 for all p ∈ P. 4. Since the uniform space X is Hausdorﬀ, we have shown at the end of steps 2 and 3 that the function f : T = S ∪ (T \ S) → X is well-deﬁned, ∞ and it is the pointwise limit on T of a subsequence {fjk }k=1 of the original ∞ sequence {fj}j=1. It follows from Lemma 1(d) that, given p ∈ P and ε0 > 0,

|f(T )|p ≤ lim inf |fj (T )|p ≤ lim inf Vε ,p(fj , T )+2ε0 k→∞ k k→∞ 0 k

≤ lim sup Vε0 ,p(fj, T )+2ε0 < ∞, j→∞

19 and so, f is a bounded function on T (i.e., f ∈ B(T ; X)). Now, we prove that f is regulated on T . Given τ ∈ T , which is a left limit point for T , let us show that dp(f(s), f(t)) → 0 as T ∋ s, t → τ − 0 for all p ∈ P (similar arguments apply in the case when τ ′ ∈ T is a right limit point for T ). Given p ∈ P, this is equivalent to showing that for every η > 0 there is δ = δ(η,p) > 0 such that dp(f(s), f(t)) ≤ 7η for all s, t ∈ T ∩ (τ − δ, τ) with s < t. Recall that the (ﬁnally) extracted subsequence of the original sequence {fj}, again denoted by {fj} here, satisﬁes condition (4.1), and fj → f pointwise on T .

Let p ∈ P and η > 0 be arbitrarily fixed. Since εk → 0 as k → ∞, pick + and fix natural k = k(η) such that εk ≤ η, and since ϕk,p ∈ Mon(T ; R ) + and τ ∈ T is a left limit point for T , the left limit limT ∋t→τ−0 ϕk,p(t) ∈ R exists. Hence, there is δ = δ(η,k,p) > 0 such that |ϕk,p(t) − ϕk,p(s)| ≤ η for all s, t ∈ T ∩ (τ − δ, τ). By virtue of (4.1), for any s, t ∈ T ∩ (τ − δ, τ) there is j1 = j1(η,k,s,t,p) ∈ N such that, if j ≥ j1, then the inequalities (4.4) hold. Arguing exactly the same way as in between the lines (4.4) and (4.5), 1 we find that dp(fj(s), fj(t)) ≤ 6η for all j ≥ j . Noting that fj(s) → f(s) and fj(t) → f(t) in X as j →∞, by the triangle inequality for dp, we get

|dp(fj(s), fj(t)) − dp(f(s), f(t))| ≤ dp(fj(s), f(s))+ dp(fj(t), f(t)) → 0

2 2 as j → ∞. So, there is j = j (η,s,t,p) ∈ N such that dp(f(s), f(t)) ≤ 2 1 2 dp(fj(s), fj(t))+ η for all j ≥ j . Thus, choosing j ≥ max{j , j }, we obtain dp(f(s), f(t)) ≤ 6η + η =7η. This completes the proof of Theorem 2.

Remark 2. Theorem 2 is an extension of the Helly-type selection principles from [21, Theorem 3.8] (for T = [a, b] and X = RN ) and [14, Theorem 3] (for any T ⊂ R and a metric space X). It also extends Theorem 8 from [10], in which, under the assumptions of Theorem 2, condition (2.2) is replaced by a more stringent one: Cp ≡ supj∈N Vp(fj, T ) < ∞ for all p ∈ P. In fact, by Lemma 1 (c), Vε,p(fj, T ) ≤ Cp for all j ∈ N, ε > 0 and p ∈ P, and so, (2.2) is fulﬁlled. Now, if, according to Theorem 2, a subsequence of {fj} converges pointwise on T to a function f ∈ XT , then property (3.6) implies Vp(f, T ) ≤ Cp for all p ∈ P, i.e., f ∈ BV(T ; X).

Remark 3. Condition (2.2) in Theorem 2 is necessary for the uniformly T convergent sequence {fj} ⊂ X in the following sense: if fj ⇒ f on T ,

20 T where f ∈ X is such that Vε,p(f, T ) < ∞ for all ε> 0 and p ∈ P, then, by virtue of Lemma 2(a), for all 0 <ε′ <ε and p ∈ P, we have

lim sup Vε,p(fj, T ) ≤ Vε−0,p(f, T ) ≤ Vε′,p(f, T ) < ∞. j→∞ Example 5. Condition (2.2) in Theorem 2 is not necessary for the pointwise T convergent sequence {fj} ⊂ X (cf. also Examples 4.7, 4.10 and 4.11 in [13]). This can be illustrated by the sequence {fj} from Example 4, where I = [0, 1]. We assert that if p ∈ P is such that dp(x, y) > 0, and 0 <ε

0= t0

I Supposing g ∈ X is arbitrary such that dI,p(fj,g) ≤ ε, by virtue of the deﬁnition of Vp(g,I) and (3.2), we ﬁnd

j! j!

Vp(g,I) ≥ dp(g(tk),g(sk)) ≥ dp(f(tk), f(sk))−2ε =j!·(dp(x, y)−2ε). k=1 k=1 X X By deﬁnition (2.1), Vε,p(fj,I) ≥ j!·(dp(x, y)−2ε), which proves our assertion.

Example 6. Theorem 2 is inapplicable to the sequence {fj} from Exam- ple 2, because (although fj ⇒ f ≡ Dx,y on I) limj→∞ Vε(fj,I) = ∞ for ε = kx − yk/2. The reason here is that the limit function Dx,y is not regulated if x =6 y. Nevertheless, Theorem 2 can be applied to sequences of non-regulated functions. Let {xj} and {yj} be two sequences in the uniform space X such that xj =6 yj for all j ∈ N, xj → x and yj → x in X as j → ∞ for some x ∈ X. The sequence fj = Dxj ,yj converges uniformly on I = [a, b] to the constant function f(t) ≡ x on I:

dI,p(fj, f) = max{dp(xj, x),dp(yj, x)}→ 0 as j →∞ for all p ∈ P.

Given p ∈ P and ε > 0, there is j0 = j0(p,ε) ∈ N such that dp(xj,yj) ≤ ε for all j ≥ j0, and so, by (3.12), Vε,p(fj,I) = 0 for all j ≥ j0. This implies condition (2.2):

lim sup Vε,p(fj,I) ≤ sup Vε,p(fj,I)=0. j→∞ j≥j0

21 Applying Theorem 2 and the diagonal process over expanding intervals, we get the following local version of Theorem 2:

Theorem 3. Under the assumptions of Theorem 2, suppose that condition (2.2) is replaced by (the local one)

lim sup Vε,p(fj, T ∩ [a, b]) < ∞ for all a, b ∈ T , a ≤ b, ε> 0 and p ∈ P. j→∞

Then, there is a subsequence of {fj} which converges pointwise on T to a function f ∈ Reg(T ; X) such that f ∈ B(T ∩ [a, b]; X) for all a, b ∈ T , a ≤ b.

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