%Typeset with Latex format \documentclass{article} %\usepackage{amsmath,amssymb} \usepackage{amssymb} \newcommand{\forces}{\mathrel{\|}\joinrel\mathrel{-}} \newcommand{\suc}{{\rm succ}} \newcommand{\cf}{{\rm cf}} \newcommand{\cc}{{\rm con}} \newcommand{\ccc}{{\rm con}} \newcommand{\supp}{{\rm supp}} \newcommand{\nor}{{\rm norm}} \newcommand{\norsup}{{\rm\underline{norm}}} \newcommand{\sq}{\subseteq} \newcommand{\real}{{\mathbb R}} %\newcommand{\real}{{\Re}} \newcommand{\R}{{\real}} \newcommand{\rational}{{\mathbb Q}} \newcommand{\Q}{{\rational}} \newcommand{\integer}{{\mathbb Z}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\NN}{{\cal N}} \newcommand{\nnn}{{\rm N}} \newcommand{\Sg}{{\Sigma}} \newcommand{\s}{{\sigma}} \newcommand{\h}{{\aleph}} \newcommand{\la}{{\langle}} \newcommand{\ra}{{\rangle}} \newcommand{\restr}{{\hbox{$\,|\grave{}\,$}}} \newcommand{\srestr}{{\hbox{${\scriptstyle\,|\grave{}\,}$}}} \newcommand{\lin}{{\rm Lin_{\rational}}} \newcommand{\A}{{\rm A}} \newcommand{\add}{{\rm Add}} \newcommand{\M}{{\cal M}} \newcommand{\F}{{\cal F}} \newcommand{\G}{{\cal G}} \newcommand{\C}{{\rm C}} \newcommand{\B}{{\cal B}} \newcommand{\T}{{\cal T}} \newcommand{\D}{{\cal D}} \newcommand{\E}{{\cal E}} \newcommand{\e}{{\emptyset}} \newcommand{\ep}{{\varepsilon}} \newcommand{\g}{{\gamma}} \renewcommand{\d}{{\delta}} \renewcommand{\H}{{\cal H}} \renewcommand{\P}{{\cal P}} \newcommand{\ext}{{\rm Ext}} \newcommand{\conn}{{\rm Conn}} \newcommand{\Ccont}{{\cal C_{< \cont}}} \newcommand{\ccont}{{\rm C_{< \cont}}} \newcommand{\ad}{{\rm AD}} \newcommand{\ac}{{\rm AC}} \newcommand{\da}{{\rm D}} \newcommand{\pc}{{\rm PC}} \newcommand{\ccf}{{\rm CC}} \newcommand{\sd}{{\rm SD}} \newcommand{\sz}{{\rm SZ}} \newcommand{\CC}{{\cal CC}} \newcommand{\SZ}{{\cal SZ}} \newcommand{\const}{{\rm Const}} \newcommand{\bor}{\mbox{${\cal B}or$}} \newcommand{\bd}{{\rm bd}} \newcommand{\per}{{\rm P}} \newcommand{\szsets}{{\cal J}_{SZ}} \newcommand{\szshift}{{\cal SZ}_{shift}} % characteristic function \newcommand{\charf}[1]{\mbox{\raise.48ex\hbox{$\chi$}$_{#1}$}} %continuum %\newcommand{\cont}{{\bf c}} \newcommand{\cont}{{\mathfrak c}} \newcommand{\cant}{{\mathfrak C}} \def\dom{{\rm dom}} \def\proof{\noindent {\sc Proof. }} \def\qed{\hfill\vrule height6pt width6pt depth1pt\medskip}%{\hfill$\Box$} %% Theorems, etc. \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{problem}[theorem]{Problem} \newtheorem{example}[theorem]{Example} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{Fact}[theorem]{Fact} \newcommand{\thm}[2]{\begin{theorem}$\!\!\!${\bf .} \label{#1}{\sl #2}\end{theorem}} \newcommand{\cor}[2]{\begin{corollary}\label{#1}{\sl #2}\end{corollary}} \newcommand{\prop}[2]{\begin{proposition}\label{#1}{\sl #2}\end{proposition}} \newcommand{\lem}[2]{\begin{lemma}\label{#1}{\sl #2}\end{lemma}} \newcommand{\pr}[2]{\begin{problem}\label{#1}{\rm #2}\end{problem}} \newcommand{\ex}[2]{\begin{example}\label{#1}{\sl #2}\end{example}} \newcommand{\defi}[2]{\begin{definition}\label{#1}{\rm #2}\end{definition}} \newcommand{\rem}[2]{\begin{remark}\label{#1}{\rm #2}\end{remark}} \newcommand{\fact}[2]{\begin{Fact}\label{#1}{\rm #2}\end{Fact}} \begin{document} %\baselineskip=23pt \title{The ideal of Sierpi{\'n}ski-Zygmund sets on the plane} \author{Krzysztof P\l otka\thanks{This paper was written under supervision of K. Ciesielski. The author wishes to thank him for many helpful conversations.}\\ Department of Mathematics, West Virginia University\\ Morgantown, WV 26506-6310, USA\\ kplotka@math.wvu.edu \\ and \\ Institute of Mathematics, Gda{\'n}sk University\\ Wita Stwosza 57, 80-952 Gda{\'n}sk, Poland} %\date{August 31, 2000} \maketitle \begin{abstract} We say that a set $X \sq \real^2$ is {\it Sierpi{\'n}ski-Zygmund\/} (shortly {\it SZ-set\/}) if it does not contain a partial continuous function of cardinality continuum $\cont$. We observe that the family of all such sets is $\cf(\cont)$-additive ideal. Some examples of such sets are given. We also consider {\it SZ-shiftable sets\/}, that is, sets $X \sq \real^2$ for which there exists a function $f\colon \real \to \real$ such that $f+X$ is an SZ-set. Some results are proved about SZ-shiftable sets. In particular, we show that the union of two SZ-shiftable sets does not have to be SZ-shiftable. \end{abstract} \vskip .5 in The terminology is standard and follows \cite{cie}. The symbol $\real$ stands for the set of all real numbers. The cardinality of a set $X$ we denote by $|X|$. In particular, $|\real|$ is denoted by $\cont$. Given a cardinal $\kappa$, we let $\cf(\kappa)$ denote the cofinality of $\kappa$. We say that a cardinal $\kappa$ is regular provided that $\cf(\kappa)=\kappa$. A set $M \sq \real^n$ is called {\it Marczewski measurable} if every perfect set $P$ has a perfect subset $Q$ such that $Q\sq M$ or $Q\cap M = \emptyset$. If every perfect set $P$ has a perfect subset $Q$ such that $Q\cap M = \emptyset$, then $M$ is called {\it Marczewski null}. We consider only real-valued functions unless stated otherwise. No distinction is made between a function and its graph. For any planar set $Y$, we denote its $x$-projection by $\dom(Y)$. For any two partial real functions $f,g$ we write $f+g$, $f-g$ for the sum and difference functions defined on $\dom(f) \cap \dom(g)$. The class of all functions from a set $X$ into a set $Y$ is denoted by $Y^X$. We write $f|A$ for the restriction of $f \in Y^X$ to the set $A \sq X$. For any function $g \in \real^X $, any family of functions $F \sq \real^X$, and any set $A \sq X\times\real$ we define $g+F =\{g+f \colon f\in F\}$ and $g+A =\{\la x,g(x)+y \ra \colon \la x,y \ra\in~A\}$. The image and preimage of a set $B$ under the function $h$ are denoted by $h[B]$ and $h^{-1}[B]$, respectively. Let us recall that a function $f\colon \real\to\real$ is {\it Sierpi{\'n}ski-Zygmund\/} ($f\in\sz$) if for every set $X \sq \real$ of cardinality continuum $\cont$, $f|X$ is discontinuous. This definition is generalized onto subsets of $\real^2$. (See~\cite{kplotka}.) \defi{defsz-set1}{A set $X \subseteq \real^2$ is called {\it Sierpi{\'n}ski-Zygmund\/} set (shortly {\it SZ-set\/}), if for every partial real continuous function $f$ we have $|f \cap X| < {\cont}$.} We denote the family of all SZ-sets by $\szsets$. Since every Sierpi{\'n}ski-Zygmund function is also an SZ-set we have that the family $\szsets$ is not empty. The next fact follows directly from the definition. \fact{p2-fact1}{$\szsets$ is a $\cf(\cont)$-additive ideal.} \proof It is obvious that $\szsets$ is closed under the operation of taking subsets. We will show that $\szsets$ is $\cf(\cont)$-additive. Take a $\kappa < \cf(\cont)$. Let $\{X_\xi \colon \xi < \kappa\} \sq \szsets$ and $f\sq \bigcup_{\xi < \kappa} X_\xi$ be a partial continuous function. Since $X_\xi$ is SZ-set, we have that $|f \cap X_\xi| < \cont$ for each $\xi < \kappa$. Consequently, $|f \cap \bigcup_{\xi < \kappa} X_\xi| = |\bigcup_{\xi < \kappa}(f \cap X_\xi)| < \cont$. \qed The question that one could ask here is how ``big'' an SZ-set can be. An example of the SZ-set that can be considered ``big'' in some sense is given in~\cite{kplotka}. \lem{p2-fact2}{\mbox{\rm \cite[Lemma 19]{kplotka} } There exists an SZ-set $X \subseteq \real^2$ such that $|\real \setminus X_x|<{\cont}$ for every $x \in \real$, where $X_x=\{y \in \real \colon \la x,y \ra \in X\}$.} Observe that the complement of every vertical section of the set $X$ has size less than $\cont$. In particular, if MA holds then every vertical section is residual in $\real$. Moreover, under CH, the complement of every vertical section of $X$ is countable. It turns out that the existence of such SZ-set (i.e., with co-countable vertical sections) is equivalent to CH. We state \prop{p2-prop1}{ {\rm CH} is equivalent to the existence of an SZ-set $X \sq \real^2$ with the following property $$|\real \setminus X_x|\le\omega \mbox{ for every } x \in \real .$$} \proof The existence of the desired set under the assumption of CH follows from the previous discussion. So we need to prove the opposite implication. Assume, by the way of contradiction, that the desired set $X$ exists and CH does not hold, e.g. $\cont > \omega_1$. Since $X$ is an $\sz$-set we get \begin{description} \item[$(\ast)$] $X^y=\{x \in \real \colon \la x,y\ra \in X\}$ has cardinality less than $\cont$ for every $y\in \real$. \end{description} We claim that there exists an $A\in[\real]^{\omega_1}$ such that $|\bigcup_{y\in A}X^y| < \cont$. The following two cases are possible. \noindent {\bf Case 1.} There exists a $\kappa < \cont$ such that $Z_\kappa=\{y\colon |X^y|=\kappa\}$ is uncountable. Then we choose $A\in [Z_\kappa]^{\omega_1}$. Obviously, $|\bigcup_{y\in A}X^y| =\kappa \omega_1 < \cont$. \noindent {\bf Case 2.} $|Z_\kappa|\le \omega$ for every cardinal $\kappa < \cont$. Put $Z=\{ |X^y| \colon y\in\real\}$ and observe that $\real=\bigcup_{\kappa\in Z}Z_\kappa$. It follows from $(\ast)$ that if $\kappa\in Z$ then $\kappa <\cont$. Consequently, since the union of less than continuum many countable sets has size less than continuum, we conclude that $|Z|=\cont$. Let $\lambda$ be the $\omega_1$-st element of $Z$. We define $A=\{y\colon |X^y|<\lambda\}$. Clearly, $|\bigcup_{y\in A}X^y| =|\bigcup_{\kappa < \lambda} Z_\kappa|\le \lambda \omega < \cont$. Now choose an $x\in\real\setminus\bigcup_{y\in A}X^y$ and notice that $(\{x\}\times A) \cap X=\e$. So $A\sq \real \setminus X_x$. This is in contradiction with the fact that every vertical section of $X$ is co-countable. \qed It is worth remarking here that SZ-sets with the Baire property or measurable are ``small.'' It means that every measurable SZ-set has measure zero and every SZ-set with the Baire property is meager. This follows from Fubini Theorem and Kuratowski-Ulam Theorem, respectively. But do such ``small'' SZ-sets exist? The answer is positive. It is easy to construct a Sierpi{\' n}ski-Zygmund function (so also an SZ-set) contained in $\real \times \cant$, whose domain is the whole real line. $\cant$ is the standard linear Cantor set. Observe also that there are ``big'' SZ-sets in terms of outer measure. The set $X$ from Lemma~\ref{p2-fact2} is of full outer measure. To see this, choose a closed set $F \sq \real^2\setminus X$. Based on the properties of $X$ we conclude that every vertical section of $F$ is countable. Hence $F$ is of measure zero. This proves that $X$ is of full outer measure. The above discussion states that ``good'' SZ-sets (in terms of measure or Baire property) are ``small''. However, we have the following \rem{szmarcz}{There exists an SZ-set which is Marczewski measurable but not Marczewski null.} \proof We claim that the set $X$ from Lemma~\ref{p2-fact2} is the desired set. Let us see why $X$ is Marczewski measurable but not Marczewski null. Fix a perfect set $P \sq \real^2$. There are two possible cases. Either some vertical section $P_a$ of $P$ is perfect, or all vertical sections are countable. In the first case, there is a $Q \sq \{a\} \times P_a$ completely contained in $X$, because the complement of every vertical section of $X$ has cardinality less than $\cont$. In the second case, we can find a partial continuous function $f \sq P$ defined on a perfect set. To see this consider a function $g\colon \dom(P) \to \real $ defined by $g(x)=\sup(P_x \cap (-\infty ,0])$. The function $g$ is upper semi-continuous so also of Baire class one. Thus, $g$ contains a continuous function defined on a perfect set. (See~\cite{oxtoby}.) Since $|f \cap X| < \cont$, the restriction of $f$ to some perfect subset $R$ of $\dom(f)$ is disjoint with $X$. Note that $f|R$ is a perfect set. Thus $P$ contains a perfect subset disjoint with $X$. It is obvious that $X$ contains a perfect set (every vertical section contains a perfect set). So $X$ is not Marczewski null. This completes the proof of our remark. \qed Another interesting observation is that the property of being an SZ-set is not preserved under the homeomorphic images. It is easy to see that any vertical line is an SZ-set, but after a rotation, for example about $\frac{\pi}{4}$, it is not an SZ-set any more. However, if $h \colon \real^2 \to \real^2$ is a homeomorphism preserving vertical lines then $h[X]$ is an SZ-set for every $X \in \szsets$. \fact{p2-fact4}{Let $h \colon \real^2 \to \real^2$ be an homeomorphism such that $h[L]$ is a vertical line for every vertical line $L$. Then $h\{\szsets\}=\{h[X] \colon X\in\szsets \}=\szsets$.} \proof First we show the inclusion $h\{\szsets\} \sq \szsets$. It is easy to see that if $f \colon A \to \real$ is a partial continuous function then $h^{-1}[f] \colon A \to \real$ is also continuous. This implies that for every $X \in \szsets$, $h[X]$ is also in $\szsets$ since $h[X]\cap f=h[X\cap h^{-1}[f]]$. Now to show the other inclusion, let us fix a $Y \in \szsets$. Note that $h^{-1}$ also preserves all vertical lines. Thus, from the first part of the proof, $X=h^{-1}[Y]~\in~\szsets$. Hence $Y=h[X]~\in~h\{\szsets\}$. \qed As we mentioned at the beginning of this paper, the concept of Sierpi{\'n}ski-Zygmund sets is a generalization of the concept of Sierpi{\'n}ski-Zygmund functions. One of the questions related to the family $\sz$ of Sierpi{\'n}ski-Zygmund functions is for how ``big'' families $F\sq \real^\real$ we can find a function $g\in\real^\real$ such that $g+F \sq \sz$. (See e.g. \cite{cie-nat}.) Similar question can be asked in the case of Sierpi{\'n}ski-Zygmund sets. This leads to the following definition. \defi{sz-shift}{A set $X \subseteq \real^2$ is called {\it $\sz$-shiftable\/}, if there exists a function $f \colon \real \to \real$ such that $f+X$ is SZ-set.} We denote the family of all SZ-shiftable sets by $\szshift$. Obviously $\szsets \sq \szshift$, so $\szshift$ is not empty. \lem{p2-lem1}{Let $X\sq\real^2$. If for all $x \in \real$ and $A \in [\real]^{<\cont}$ there exists an $a\in \real$ such that $(a+A)\cap X_x=\emptyset$, then $A$ is SZ-shiftable.} \proof Let $\la x_{\alpha}: \alpha < \cont \ra $ and $\la f_{\alpha}: \alpha < \cont \ra $ be the sequences of all real numbers and all continuous functions defined on a $G_{\delta}$ subset of $\real$, respectively. We will define a function $f \colon \real \to \real $ which shifts $X$ into $\szsets$, using transfinite induction. For every $\alpha < \cont$ we choose $f(x_{\alpha}) \in \real$ with the property that $(f(x_{\alpha})+X_{x_{\alpha}}) \cap \{f_{\xi}(x_{\alpha}) \colon \xi < \alpha \} =\emptyset$. Such a choice is possible because of the assumptions on $X$. It is easy to see that $\dom \left((f+X) \cap f_{\beta} \right ) \sq \{x_{\xi} \colon \xi < \beta \}$ for each $\beta < \cont$. Thus $f+X \in \szsets$. \hfill \qed Recall that under Martin's Axiom (MA) the union of less than $\cont$ meager sets is meager. Suppose that $A \in[\real]^{<\cont}$ and $B\sq \real$ is meager. Then the set $B-A=\bigcup_{x\in A}(B-x)$ is meager as a union of less than $\cont$ meager sets. Now, if we choose an $a\notin B-A$ then $(a+A)\cap B=\emptyset$. Notice that the same argument can be repeated for the sets of measure zero. The above discussion and Lemma~\ref{p2-lem1} immediately imply \cor{p2-cor1}{{\rm (MA)} If each vertical section of a set $X \sq \real^2$ is meager or of measure zero, then $X \in \szshift$.} It may also be of interest to determine whether $\szshift$ is closed under the union operation. Fact~\ref{p2-fact1} states, in particular, that the union of two SZ-sets is also an SZ-set. Thus, the natural question that appears here is whether the same is true for SZ-shiftable sets. It turns out not to be the case. \ex{p2-ex1}{There exist $A_1, A_2 \in \szshift$ such that $A_1 \cup A_2=\real^2 \not\in \szshift$.} \proof Put $A_1$ to be the set $X$ from Lemma~\ref{p2-fact2} and $A_2$ to be its complement. Based on Lemma~\ref{p2-lem1} $A_2$ is SZ-shiftable. Next, notice that $A_1 \in \szsets \sq \szshift$. Finally, $A_1 \cup A_2=\real^2$ and obviously $\real^2$ is not in $\szshift$. \qed Before we finish let us make a comment about \cite[Theorem 2 (1)]{kplotka} which says: MA implies that for every finite family $F$ of real functions there exists an almost continuous function $g$ (each open subset of $\real^2$ containing the graph of $g$ contains also the graph of a continuous function) such that $g+f$ is Sierpi{\'n}ski-Zygmund for every $f\in F$. Note that this result can be expressed using the notion of SZ-sets. Under MA the following holds: \begin{description} \item {\emph{\quad\quad If, for some fixed $n \in \omega$, every vertical section of the set $X \sq \real^2$ has at most $n$ elements then there exists an almost continuous function $f\colon \real \to \real$ such that $f+X \in \szsets$.}} \end{description} We generalize the above result. \thm{p2-thm1}{{\rm (MA)} If every vertical section of the set $X \sq \real^2$ is finite then there exists an almost continuous function $f\colon \real \to \real$ such that $f+X \in \szsets$.} Before we prove the theorem we need to cite some lemmas and recall some properties. First let us observe that a function $f\colon \real \to \real$ is almost continuous if and only if it intersects every {\em blocking set\/}, i.e., a closed set $K \sq \real^2$ which meets every continuous function from $\real$ to $\real$ and is disjoint with at least one function from $\real^{\real}$. Next we give some definitions needed to state the lemmas. (See \cite{kplotka}.) For $X \sq \real$ by $\ccont(X)$ we denote the family of all functions $f \colon X \to \real $ which can be represented as a union of less than $\cont$-many partial continuous functions. The symbol $\sz(X)$ denotes the family of all partial Sierpi{\'n}ski-Zygmund functions defined on $X$. Let $A \sq \real$ be everywhere of second category, that is $A \cap I $ is of second category for every nontrivial interval $I$. We define ${\cal F}_A$ as a family of all $F\subseteq \real^{\real}$ whose union $\bigcup F$ contains no function from $\ccont(A \cap B)$ for any Borel set $B$ of second category. \lem{}{{\rm \cite[Lemma 12]{kplotka}} {\rm (MA)} Let $F \in {\cal F}_A$ be a family such that $|F| \le \cont $. There exists a $g \in \sz(A) $ such that every extension $\bar{g} \colon \real \to \real$ of $g$ is almost continuous and $g+F \subseteq \sz(A) $.} A slight modification of the proof of the above lemma gives a little stronger result. (See \cite[Lemma 2.2.1]{kplotka_phd}.) \lem{lemphd1}{{\rm (MA)} Let $F \in {\cal F}_A$ be a family such that $|F| \le \cont$. There exists a $g \in \sz(A) $ such that $g+F \subseteq \sz(A)$ and for every blocking set $B\sq\real^2$ there is a non-empty open interval $I_B\sq \dom(B)$ with the property that $\dom(B\cap g)$ is dense in $I_B$.} \lem{lemphd2}{{\rm \cite[Lemma 13]{kplotka}} {\rm (MA)} Let $\{ f_i \}_1^n \subseteq \real^{\real}$, $n=1,2, \dots $. There exists $\{ f_i^{\prime} \}_1^n \in {\cal F}_A$ such that $f_i | A_i \in \ccont(A_i)$, where $A_i=\{x \colon f_i(x) \ne f_i^{\prime}(x) \}$.} Note that Lemmas~\ref{lemphd1} and ~\ref{lemphd2} imply the following. \begin{itemize} \item[($\star$)] {\rm (MA)} Assume that $F \subseteq \real^{\real}$ is finite and $A \sq \real $ is everywhere of second category. Then there exists a function $g\colon A \to \real$ such that $g+ F \sq \sz(A)$ and $\dom(g \cap B)$ is dense in some non-empty open interval $I_B$ for every blocking set $B$. \end{itemize} \proof Let us consider the partition $\{H_n \colon n \in \omega\}$ of $\real$, where $H_n$ is defined by $H_n=\{x \in \real \colon |X_x|=n\}$. Let $G_n\sq\real$ be a maximal open set such that $H_n$ is everywhere of second category in $G_n$. Such a set can be easily constructed. Simply define $G_n$ as the interior of the set $\real\setminus\bigcup_{I\in {\cal I}_n}I$, where ${\cal I}_n$ is the set of all open intervals in which $H_n$ is meager. We claim that for every $n<\omega$, there exists a function $g_n\colon (G_n\cap H_n)\to \real$ such that $g_n + X=\{\la x,g_n(x)+y\ra\colon x\in (G_n\cap H_n), \; \la x,y\ra\in X\} \in\szsets$ and $\bigcup_{n<\omega}g_n$ intersects every blocking set $B$. First observe that this claim implies the conclusion of the theorem. Put $g\colon\real\to\real$ to be an extension of $\bigcup_{n<\omega} g_n$ such that $[g|(\real\setminus\bigcup_{n<\omega}G_n\cap H_n)]+X$ is an $\sz$-set. This extension exists based on Corollary~\ref{p2-cor1}. Thus, $g+X$ is the union of countable many $\sz$-sets. Consequently, $g+X \in\szsets$. Clearly, $g$ intersects every blocking set, so $g$ is almost continuous. To complete the proof we need to show the above claim. Fix an $n<\omega$ and put $A_n=(G_n\cap H_n)\cup\bigcup_{I\in {\cal I}_n}I$. The set $A_n$ is everywhere of second category. Notice also that the part of $X$ contained in $(G_n\cap H_n)\times \real$ can be covered by $n$ functions $f_1, \dots , f_n$ from $\real$ to $\real$. So, by ($\star$), there exists a function $g_n^{\prime}\colon A_n \to \real$ such that $g_n^{\prime} + \{f_1 ,\dots ,f_n \} \sq\sz(A_n)$ and $\dom(g_n^{\prime}\cap B)$ is dense in some non-empty open interval $I_B$ for every blocking set $B$. Thus, if we define $g_n=g_n^{\prime}|(G_n\cap H_n)$ then $g_n + X\in\szsets$. What remains to prove is that $\bigcup_{n<\omega}g_n$ intersects every blocking set $B$. Notice that $I_B \cap G_n \ne \e$ for some $n$. Thus, $g_n \cap B \ne \e$. Consequently, $\e\not=B\cap\bigcup_{n<\omega}g_n\sq B\cap g$. This finishes the proof. \qed \begin{thebibliography}{99} \bibitem{balcienat} M. Balcerzak, K. Ciesielski, T. Natkaniec, {\emph {Sierpi{\'n}ski-Zygmund functions that are Darboux, almost continuous, or have a perfect road}}, Arch. Math. Logic {\bf 37} (1997), 29--35. \bibitem{cie} K. Ciesielski, {\emph {Set Theory for the Working Mathematician}}, London Math. Soc. Student Texts {\bf 39}, Cambridge Univ. Press 1997. \bibitem{cie-nat} K. Ciesielski, T. Natkaniec, {\emph {Algebraic properties of the class of Sierpi{\'n}ski-Zygmund functions}}, Topology Appl. {\bf 79} (1997), 75--99. \bibitem{kk} K. Kuratowski, {\it Topologie I}, Warszawa 1958. \bibitem{natkaniec} T. 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