Almost continuous Sierpiński–Zygmund functions under different set-theoretical assumptions

A function f:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\mathbb {R}\rightarrow \mathbb {R}$$\end{document} is: almost continuous in the sense of Stallings, f∈AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{AC}$$\end{document}, if each open set G⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\subset \mathbb {R}^2$$\end{document} containing the graph of f contains also the graph of a continuous function g:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:\mathbb {R}\rightarrow \mathbb {R}$$\end{document}; Sierpiński–Zygmund, f∈SZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{SZ}$$\end{document} (or, more generally, f∈SZ(Bor)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{SZ}(\textrm{Bor})$$\end{document}), provided its restriction f↾M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\restriction M$$\end{document} is discontinuous (not Borel, respectively) for any M⊂R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M\subset \mathbb {R}$$\end{document} of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function f:R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\mathbb {R}\rightarrow \mathbb {R}$$\end{document} (i.e., an f∈SZ∩AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{SZ}\cap \textrm{AC}$$\end{document}) cannot be constructed in ZFC; however, an f∈SZ∩AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{SZ}\cap \textrm{AC}$$\end{document} exists under the additional set-theoretical assumption cov(M)=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$\end{document}, that is, that R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} cannot be covered by less than c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {c}$$\end{document}-many meager sets. The primary purpose of this paper is to show that the existence of an f∈SZ∩AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{SZ}\cap \textrm{AC}$$\end{document} is also consistent with ZFC plus the negation of cov(M)=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$\end{document}. More precisely, we show that it is consistent with ZFC+cov(M)<c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{cov}\,}}(\mathcal {M})<\mathfrak {c}$$\end{document} (follows from the assumption that non(N)<cov(N)=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{non}\,}}(\mathcal {N})<{{\,\textrm{cov}\,}}(\mathcal {N})=\mathfrak {c}$$\end{document}) that there is an f∈SZ(Bor)∩AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \textrm{SZ}(\textrm{Bor})\cap \textrm{AC}$$\end{document} and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either cov(M)=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$\end{document} or non(N)<cov(N)=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{non}\,}}(\mathcal {N})<{{\,\textrm{cov}\,}}(\mathcal {N})=\mathfrak {c}$$\end{document}, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated.


Question 1.1 Is the condition cov(M) = c equivalent to the statement "there exists a Darboux Sierpiński-Zygmund function"?
Ciesielski and Seoane-Sepúlveda constructed in the survey [18] an example of Darboux Sierpiński-Zygmund function f : R → R under an additional set-theoretical cov(N ) = c. Since there are models of ZFC in which cov(N ) = c > cov(M), this solves Question 1.1 in the negative.
As any almost continuous function is Darboux (i.e., AC ⊂ D, see e.g. [34]) it is natural to examine also the following variant of Question 1.1: Of course, the sufficiency of the condition cov(M) = c was proved in the previously mentioned article [4]. One of the goals of this article is to answer this question, in negative, by showing that non(N ) < cov(N ) = c (under which cov(M) < c) also implies that SZ(Bor) ∩ AC = ∅. This is proved in Sect. 2.
Additionally, we show how other results concerning class SZ ∩ AC, often known under assumption cov(M) = c, can be deduced from the opposite assumption non(N ) < cov(N ) = c or its strengthening. Specifically, in Sect. 3 we study the lineability of SZ(Bor) ∩ AC, in Sect. 4 its additivity coefficient, while in Sect. 5 the existence of functions in different subclasses of SZ(Bor) ∩ AC.

Maps in SZ(Bor) ∩ AC when non(N ) < cov(N ) = c
In what follows any closed set E ⊂ R 2 which x-projection is a non-degenerate interval is called a blocking set. It is well-known that any f : R → R which meets all blocking sets is almost continuous, see e.g. [28]. (Compare also [17, lemma 5.1] and related history. ) We start with the following minor modification of [ Proof Let E n = E ∩([−n, n]×[−n, n]) for n ∈ N. Then each E n is compact, hence dom(E n ) is closed, and dom(E) = n∈N dom(E n ), so for some n ∈ N the set dom(E n ) contains a non-degenerate interval J , hence dom(E n ) / ∈ M ∪ N . Now, the functionĥ : dom(E n ) → R defined byĥ(x) := max{y ∈ [−n, n]: x, y ∈ E n } is Borel (in fact, it is upper semicontinuous) and it is contained in E. Then any Borel extension h : R → R ofĥ is as we need.
To show that non(N ) < cov(N ) = c implies SZ(Bor) ∩ AC = ∅ we will also use the following lemma. The property of the set S constructed there means that S is dense in the density topology on R. Note that Tall (see [35, theorem 4.15]) was the first who noticed that in the model of ZFC obtained by adding ω 2 random reals to the model for ZFC+CH the density of the density topology is equal to ω 1 < c. Proof To see that the main part of the lemma holds, fix any set A ⊂ R with |A| = non(N ) and A / ∈ N . Let E ⊂ R be a measurable hull of A. Clearly |E| = c, so there exists a Borel isomorphism ϕ : R → E which maps null sets in R onto null subsets of E, see e.g. [11, remark after theorem 4.12]. One can easily verify that S := ϕ −1 (A) is as needed: clearly To see the additional part of the lemma, fix an E as in the statement. First notice that (•) holds if we additionally assume that E ⊂ N . Indeed, under such assumption, there exists an is the union of less than cov(N )-many null sets, a contradiction. The set Finally, if we do not assume that E ⊂ N , let H ∈ B be a measurable hull of R\ E. If H ∈ N , then D := ∅ satisfies (•). So, we can assume that H / ∈ N . Notice thatĒ := {E ∩ H : E ∈ E} is contained in N and let D 0 be as above for the familyĒ.
The following lemma is the main tool of this paper and shows that if F ∈ R R extends f from the lemma, then F ∈ AC. This result will be used several times in this paper, when the functions of interest will be constructed by transfinite induction containing the sequence G α , f α , D α : α < c satisfying the assumptions (M), (N), (D), and (F). For the case when cov(M) = c, Lemma 2.3 is a close variation of [12, lemma 2.5]. We include its proof also for this case to emphasize the similarities and differences with its proof in the case when non(N ) < cov(N ) = c.

Lemma 2.3
Assume that either cov(M) = c or non(N ) < cov(N ) = c and let Bor = {h α : α < c}. Assume also that the sequence G α , f α , D α : α < c satisfies the following properties for every α < c: If f := α<c f α , then K ∩ f = ∅ for every blocking set K .
Before we prove the lemma we like to show how it implies the following theorem, the main result of this section. • if non(N ) < cov(N ) = c then |D α | ≤ non(N ) and the additional property of the set D α is ensured by Lemma 2.2 applied to the family E : So, let f : E → R be as in Lemma 2.3 and g : R → R be any map in SZ(Bor). Then F := g (R\E) ∪ f is as needed. Indeed, F ∈ AC is ensured by Lemma 2.3. To see that F ∈ SZ(Bor) fix an h ∈ Bor and choose a β < c such that h = h β . By (F) and (D), for any α > β and has cardinality less than c. Also, |g ∩ h| < c, giving desired |F ∩ h| < c.
Proof of Lemma 2.3 Fix a blocking set K ⊂ R 2 . It is enough to show that there exists an To see this let I = M when cov(M) = c and I = N otherwise. By Lemma 2.1, dom(h ξ ∩ K ) ∈ B\I for some ξ < c. 1 Let α < c be the first ordinal with this property. Then • dom(h α ∩ K ) ∈ B\I and dom(h β ∩ K ) ∈ I for each β < α.
Consider two cases. non(N ) < cov(N ) = c: By (1) and part of (D) applied to the set B : so, by (D), there is an x ∈ D α ∩ J ∩ Z α . In particular, by (F), x, f (x) = x, h α (x) . So, to finish the proof it is enough to show that x, h α (x) ∈ K . This would be obvious if we could ensure that x belongs also to dom(h α ∩ K ); however, it is possible that dom(h α ∩ K ) ∩ D α = ∅, what prevents such choice of x. So, we will use another argument to show that x, h α (x) ∈ K .
To see this, notice that, by the property (2), for every n < ω there exists an In the case cov(M) = c, the key property in the proof of Lemma 2.3 is that the first part of • implies existence of a non-empty open interval J 0 for which J 0 \dom(h α ∩ K ) ∈ I. Of course, there is no such property for I = N and this is the main reason why a similar proof does not work in this case. in lineability, introduced by Gurariȋ, has been a rapidly developing trend in both recent real and functional analysis, see e.g. [2,3], or [9].
In 2015, Płotka, assuming CH, proved that the family AC∩SZ is c + -lineable [31]. (Clearly, such a result cannot be proved in ZFC, as it is consistent that AC∩SZ = ∅.) It was noticed, in a 2017 paper [13] of Ciesielski, Gámez-Merino, Mazza, and Seoane-Sepúlveda (and repeated in a survey [18]), that the argument from [31] actually works under weaker assumption cov(M) = c. This, however, was not precise, since (as we will see below) the argument requires also the assumption that c is regular, which does not follow from cov(M) = c.
The goal of this section is to prove the following theorem, which clarifies the situation under the assumption of cov(M) = c and shows that the same result holds also under assumption non(N ) < cov(N ) = c. It is worth to mention that the class AC (as well as any of its subclass discussed in Sect. 5) is 2 c -lineable, see [1] and [9]. However, 2 c -lineability for the class SZ(Bor) is undecidable in ZFC. More precisely, it is not difficult to see that SZ(Bor) is c + -lineable. Thus, if 2 c = c + (e.g., under GCH), then SZ(Bor) is 2 c -lineable. On the other hand, there are models of ZFC in which SZ is not 2 c -lineable see [20] or [18, section 3]. Even more, the assumptions of Theorem 3.1 also do not decide 2 c -lineability of AC ∩ SZ(Bor). Specifically, either of the set-theoretical assumptions of the theorem is consistent with 2 c = c + : in the model of ZFC obtained by adding ω 2 Cohen reals to a model of ZFC+GCH (where we have cov(M) = c) and, respectively, in the extension of a model of ZFC+GCH by adding ω 2 random reals (where we have non(N ) < cov(N ) = c, see next section). At the same time, if one starts with the model M of ZFC+GCH, extends it by adding ω 4 subsets of ω 1 using countable supported functions, then we obtain a model M 1 of ZFC such that in any ω 2 -cc generic extension of M 1 the family SZ is not c ++ -lineable, see [18, remark (κ) in the proof of theorem 3.3]. Hence, if we add to M 1 either ω 2 Cohen reals (to get cov(M) = c) or ω 2 random reals (to get non(N ) < cov(N ) = c), then in such extensions SZ is not 2 c -lineable.
The proof of Theorem 3.1 will easily follow from the next lemma, a variant and consequence of Lemma 2.3. Notice that the assumption G ⊂ SZ(Bor) ∪ {0} in its statement is crucial, as we prove in Theorem 4.1.

Lemma 3.2 If c is regular and either
Proof Let G = {g β : β < c} and fix a sequence h α ∈ Bor : α < c in which every h ∈ Bor appears c many times. For every α < c choose G α ⊂ R and κ such that (m) if cov(M) = c, then κ = ω and G α is residual such that h α G α is continuous; (n) if non(N ) < cov(N ) = c, then κ = non(N ) and G α = R.
By double induction on β ≤ α < c we define the sequences D α β : β ≤ α < c of pairwise disjoint subsets of R each of cardinality at most κ and f α β : β ≤ α < c of partial maps, each with domain D α β , aiming for f : For every α < c let and notice that, by our assumption on G and regularity of c, T α has a cardinality less than c.
To ensure that such f is as needed, we make sure that the following inductive conditions are satisfied for every α < c and β ≤ α: which in the case when non(N ) < cov(N ) = c has also the property that The possibility of such a construction is obvious unless non(N ) < cov(N ) = c, in which case the additional property of the set D α β can be ensured by Lemma 2.2 applied to the family To see (i) choose g = g β ∈ G and h = h δ ∈ Bor. It is enough to show that dom(( f + g E) ∩ h) ⊂ β≤γ ≤max{δ,β} D α γ as this last set has cardinality less than c. To see this inclusion, fix α > max{δ, β}, γ ≤ α, and x ∈ D α γ . We need to show that . This clearly implies that Finally, to find indicated extension F, notice that there exists a ψ ∈ R R such that ψ + G ⊂ SZ(Bor), see [15, theorem 2.1]. 3 Then F := f ∪ (ψ (R\E)) is a needed extension. Indeed, the properties (i), (ii), and the definition of ψ ensures that F + G ⊂ AC ∩ SZ(Bor). Such F cannot belong to G, since otherwise also −F ∈ G and F + (−F) / ∈ SZ(Bor).

Additivity of the family SZ(Bor) ∩ AC
For F ⊂ R R , the additivity coefficient of F is defined as (For more information on additivity see, for instance, [22].) The goal of this section is to study this coefficient for the class SZ(Bor) ∩ AC.
There is an interesting relation between add(F) and κ-lineability of F , see e.g. [19, theorem 2.4]  if F ⊂ R R is closed under non-zero scalar multiplications and add(F) > κ ≥ c, then F is κ + -lineable.
Thus, one may be tempted to provide an alternative proof of Theorem 3.1, that SZ(Bor) ∩ AC is c + -lineable, by showing that under the same set theoretical assumption we have add(SZ(Bor) ∩ AC) > c. However, this is impossible, as shown by the following result. The proof of the second inequality is a small variation of the proof of [13, theorem 2.10]. To see this, let {r ξ : ξ < c} be an enumeration of R and, for every ξ < c, define To see this notice that g = g + χ A 0 ∈ g + F. If g / ∈ SZ ∩ D we are done. So assume that g ∈ SZ ∩ D. Then g[R] contains a non-trivial interval (c, d). Take a y ∈ (c, d) and notice that A = g −1 (y) has cardinality smaller than c (as g ∈ SZ). Since c is regular, there is a ξ < c with A ⊂ A ξ . Choose an r ∈ (0, ∞)\(y − g[A ξ ]). It is enough to prove that G := g + r · χ A ξ ∈ g + F is not Darboux.
To see this, first notice that y / for every x ∈ A ξ we have G(x) = g(x) + r = y, as guaranteed by the choice of r .
On the other hand, there are p ∈ (c, y) and q ∈ (y, d) This means that G[R] is not connected, so that indeed G / ∈ D.
The main goal of this section is to prove Theorem 4.6, which gives, consistently, a lower bound for add(SZ(Bor) ∩ AC) as ω. For this, we will need some preliminaries.
Let I ∈ {M, N }. We use the symbol I <c to denote the ideal of all subsets of R that are the unions of less than c-many sets from I. We say that S ⊂ R is everywhere I <c -positive provided B ∩ S / ∈ I <c for every B ∈ B\I. We will be interested in this notion only under the assumption that cov(I) = c, in which case no B ∈ B\I belongs to I <c . Notice also that if c is a regular cardinal, then the ideal I <c is c-additive, that is, a union of less than c-many sets from I <c still belongs to I <c . Proof Let U be the maximal family of pairwise disjoint sets B ∈ B\I for which there exists h B ∈ Bor such that Notice that U is at most countable, since B\I is ccc. So, G = U is Borel. Let γ be any Borel extension of the partial function B∈U h B B. Define and observe that S is everywhere I <c -positive. Indeed, if E ∈ B\I, then either E\G / One can easily verify that S, γ , and G satisfy the statement (a). To prove statement (b), suppose that A := dom(g ∩ h) ∩ (S\G) / ∈ I <c for some h ∈ Bor and observe that then there is a B ∈ B\I such that E ∩ A / ∈ I <c for every E ∈ B\I, E ⊂ B. Indeed, let C be the maximal family of pairwise disjoint Borel sets C / ∈ I such that C ∩ A ∈ I <c , and let C 0 = C. Then C 0 ∈ B, C 0 ∩ A ∈ I <c , and B := R\C 0 is as we need. (It is easy to see that B ∈ B but also B / ∈ I by the maximality of C.) But this contradicts the maximality of the family U. Notice that S 0 = R is everywhere I <c -positive by our assumption that cov(I) = c. By applying Lemma 4.2 iteratively we can find functions γ g i , sets B g i , and a sequence S 0 ⊃ S 1 ⊃ · · · ⊃ S n so that the conditions (a) and (b) are satisfied with S i , B g i , and γ g i in place of S, B, and γ , respectively. Then S := S n is as needed.
We will also need the following fact. Proof This fact is stated in [35, paragraph above theorem 4.15] and is attributed to Professor Kenneth Kunen. However, [35] contains neither proof of this fact nor a reference to a printed source where a proof can be found. Therefore, for reader's convenience, we provide here a sketch of its proof. 4 Let M be a model of ZFC+CH and for an ordinal α let B(α) be a random real forcing on 2 α , see [7, page 99]. We will show that (d) holds in the model M ω 2 := M[h] which is an extension of M via forcing B(ω 2 ), where h ∈ 2 ω 2 is M-generic over B(ω 2 ). Recall that in this setting for every α < ω 2 there is an associated generic extension M α := M[h α] of M over B(α) and that M ω 2 is a generic extension of M α over the random real forcing B(ω 2 \α). (3) Also M α ⊂ M β for every α < β < ω 2 and we have CH in M α . First we will show that, in M ω 2 , Indeed, the assumption on A means that, in M ω 2 , there is a map ψ, a subset of 2 ω × R, such that if c ∈ 2 ω is a code for a Borel set 5ĉ ∈ B ∩ N , then ψ(c) is a real number in A\N . Since random real forcing is ccc, every pair in 2 ω × R belongs to M α for some α < ω 2 . Therefore, there is a λ < ω 2 of cofinality ω 1 such that if C λ ⊂ 2 ω is the set of all Borel codes that are in M λ , then ψ C λ belongs to M λ . But this means that, in M λ , the set B := A ∩ M λ is not in N . Now, (3) and the following property that can be found in [7, lemmas 6.3.11 and 6.3.12] • any B ⊂ R not in N is also not in N in any random real extension imply that B / ∈ N also in M ω 2 . Since CH holds in M λ , we have also |B| ≤ ω 1 , that is, B is the set that satisfies ( * ). Now, to finish the proof (d), choose a non-null set Z ⊂ R. Without loss of generality we can assume that it is bounded so that its outer measure m * (Z ) is finite.
then there is an E ∈ B disjoint with D and such that A := E ∩ Z is not in N . Then, by ( * ), there is a B ⊂ A so that |B| ≤ ω 1 and B / ∈ N . But then C := D ∪ B contradicts the maximality of d. Then for every finite family G = {g i : i < n} ⊂ R R there exists an f from E ⊂ R to R such that for every i < n Proof Let Bor = {h α : α < c}. For every α < c choose G α ⊂ R and κ such that (m) if (μ) holds, then κ = ω and G α is residual such that h α G α is continuous; (n) if (ν) holds, then κ = non(N ) and G α = R.
By induction on α < c define the sequences D α i : i < n & α < c of pairwise disjoint subsets of S each of cardinality at most κ and f α : α < c of partial maps, each from D α := i<n D α i to R, aiming for f := α<c f α . To ensure that such f is as needed, we make sure that the following inductive conditions are satisfied for every α < c and i < n: which in the case when non(N ) < cov(N ) = c has also the property that B ∩ D α i = ∅ for every B ∈ B\N such that B ∩ E ∈ N for every E in the family First choose β < c such that h = h β and notice that, by (f), for every α > β and Hence |( f + g i (E j ∩ B g j −g i )) ∩ h| < c as its x-axis projection is contained in α≤β D α j . The property (ii) follows from Lemma 2.3 when we notice that, for every i < n, κ and the sequence G α , ( f α D α i ) + (g i D α i ), D α i : α < c satisfies its assumption, so the map α<c ( f α D α i ) + (g i D α i ) contained in f intersects every blocking set K .

Theorem 4.6
Assume that c is a regular cardinal and that one of the following two assumptions holds: Then for every finite G ⊂ R R there is an F ∈ R R such that F + G ⊂ SZ(Bor) ∩ AC. In particular, add(SZ(Bor) ∩ AC) ≥ ω, hence every function f : R → R can be represented as a sum of two functions from the class SZ(Bor) ∩ AC.
Proof Recall that for every family F ⊂ R R with |F| ≤ c there exists a function g : R → R such that g + F ⊂ SZ(Bor), see the proof of Lemma 3.2. In particular, there exists a g ∈ R R be so that g + G ⊂ SZ(Bor). Let f : E → R be as in Lemma 4.5. Then map F := f ∪ (g (R\E)) is as needed.

Subclasses of SZ(Bor) ∩ AC when non(N ) < cov(N ) = c
Recently (see e.g. [6,12,18]) there has been a considerable interest in the classes SZ ∩ F , including subclasses of SZ ∩ AC, where F is one of the classes in the algebra generated by Darboux-like families of functions. Recall, that the Darboux-like families of functions usually include eight classes, of which we are interested here in AC and the following two classes defined below, see e.g. [21] of [17]. 6 PR of all functions f : R → R with perfect road, that is, such that for every x ∈ R there exists a perfect P ⊂ R containing x such that x is a bilateral limit point of P (i.e., with x being a limit point of (−∞, x) ∩ P and of (x, ∞) ∩ P) and that f P is continuous at x. CIVP of all functions f : R → R with Cantor Intermediate Value Property, that is, such that for all p, q ∈ R with f ( p) = f (q) and for every perfect set K between f ( p) and f (q), there exists a perfect set P between p and q such that f [P] ⊂ K .
The main result of this section is as follows.
Such a sequence can be constructed by an easy transfinite induction on α < c, where the additional property of the set D α 0 in the case when non(N ) < cov(N ) = c can be ensured by Lemma 2.2 applied to the family E := E α ∪ {{x}: x ∈ β<α D β }. We claim that f := α<c f α is as needed.
Indeed, the sequence G α , f α D α 0 , D α 0 : α < c satisfies the assumptions of Then f ∅ ∈ SZ(Bor) ∩ AC\PR, where f ∅ / ∈ PR is justified by (C). In the remainder of this proof, fix an M ∈ M ∩ N defined as M := G, where G := {P p,q ⊂ ( p, q) : p < q & p, q ∈ Q} is a family of pairwise disjoint compact perfect sets from N . This is the setM defined at the beginning of section 4 in [12].
In [12, lemma 4.1] it is constructed 7 a g 0 : M → R such that (R) |g 0 ∩ h| < c for every h ∈ Bor; (S) for every perfect K ⊂ R and a < b there exists a perfect P ⊂ M ∩ (a, b) such that g 0 [P] ⊂ K .
Similarly, for C denoting the classic Cantor ternary set in [0, 1], in [12, lemma 4.3] it is constructed 8 a g 1 : M → R such that (α) |g 1 ∩ h| < c for every h ∈ Bor; (β) g 1 [M] ∩ C = ∅; (γ ) for any s, t ∈ R 2 there is a perfect set P ⊂ M ∪ {s} having s as a bilateral limit point and such that lim x→s, x∈P g 1 (x) = t.
In [12] it is also shown that, besides classes listed in Theorem 5.1, also the following classes are nonempty under the assumption that cov(M) = c: In this context the following is a natural question.

Problem 5.3
Does non(N ) < cov(N ) = c imply that that the above six classes are nonempty? What about when the class SZ is replaced with SZ(Bor)?
It is our believe that the answer to this question, in its both versions, is positive. The key of proving this result should be to first prove that, under our assumption, the classes SZ(Bor) ∩ (D\Conn) and SZ(Bor) ∩ (Conn\AC) are nonempty. Then technic developed in [12] and used in the proof of Theorem 5.1 should allow to refine such results to the aforementioned six classes.
It might be also interesting to consider the same problem under a weaker set theoretical assumption of just cov(N ) = c. A positive answer to this version, for at least the first three classes, is suggested by the fact that already cov(N ) = c implies SZ(Bor) ∩ D = ∅, as shown in [18, theorem 4.4].
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