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%\controldates{6-OCT-1997,6-OCT-1997,6-OCT-1997,28-OCT-1997}
 
\documentclass{proc-l}
%\issueinfo{126}{02}{February}{1998}
\pagespan{471}{479}
\PII{S 0002-9939(98)04098-2}
\def\copyrightyear{1998}

\usepackage{amssymb}






\newcommand\la{\langle}
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\newcommand\restr[1]{\restriction#1}

\newcommand\A{{\mathfrak A}}
\newcommand\Aa{{\mathcal A}}
\newcommand\F{{\mathcal F}}
\newcommand\G{{\mathfrak G}}
\newcommand\J{{\mathcal J}}
\newcommand\K{{\mathcal K}}
\newcommand\cof{\operatorname{cof}}
\newcommand\bclass{\operatorname{BD}}
\newcommand\bfun{\operatorname{Bd}}
\newcommand\dom{\operatorname{dom}}
\newcommand\reals{{\mathbb R}}
\newcommand\rationals{{\mathbb Q}}
\newcommand\mathN{{\mathbb N}}
\newcommand\e{{\varepsilon}}

\newcommand\charf[1]{{\mathord{\raisebox{.2em}{$\chi$}}_{#1}}}

\newcommand\add{{\operatorname A}}
\newcommand\addb{{\add_{\textup{b}}}}

\newcommand\AC{{\operatorname{AC}}}
\newcommand\Conn{{\operatorname{Conn}}}
\newcommand\Darb{{\operatorname D}}
\newcommand\Ext{{\operatorname{Ext}}}
\newcommand\PC[1]{{\operatorname{PC}_{#1}}}
\newcommand\PR{{\operatorname{PR}}}
\newcommand\U[1]{{{\mathcal U}_{#1}}}

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\begin{document}
\title[Cardinal invariants]{Cardinal 
invariants concerning \\ bounded families of 
 extendable \\ and almost continuous functions}
\author{Krzysztof Ciesielski}
\address{Department of Mathematics, West
 Virginia University, Morgantown, West Virginia 26506--6310}
\email{kcies@wvnvms.wvnet.edu}
\author{Aleksander Maliszewski}
\address{Department of Mathematics, Pedagogical
 University, Arciszewskiego 22, 76--200 \linebreak  S\l{}upsk, Poland}
\email{wspb05@pltumk11.bitnet}
\commby{J. Marshall Ash}
\keywords{Peripheral continuity, almost
 continuity, connectivity, extendability}\date{March 28, 1996 and, in revised form, August 11, 1996}
\subjclass{Primary 26A21; Secondary 54C08}
\thanks{This work was partially supported by NSF Cooperative Research Grant
INT-9600548}
\begin{abstract}
 In this paper we introduce and examine a cardinal
 invariant~ $\operatorname{A}_{\textup{b}}$ closely connected
 to the addition of bounded functions from $\mathbb{R}$ to~$\mathbb{R}$. It
 is analogous to the invariant~$\operatorname{A}$ defined earlier for arbitrary
 functions by T.~Natkaniec. In particular, it is proved that each
 bounded function can be written as the sum of two bounded almost
 continuous functions, and an example is given that there is a
 bounded function which cannot be expressed as the sum of two
 bounded extendable functions.
\end{abstract}

\maketitle

\section{Preliminaries}
 We will use the following terminology and notation.
 The letters $\mathN$, $\rationals$, and~$\reals$ denote the set
 of positive integers, the set of rationals, and the real line,
 respectively. Functions will be identified with their graphs.
 The family of all
 functions from a set $X$ into~$Y$ will be denoted by $Y^X$. The
 symbol $|X|$ will stand for the cardinality of a set~$X$. We
 consider cardinals as ordinals not in one-to-one correspondence
 with the smaller ordinals. For a cardinal number~$\kappa$ we will
 write $\cof(\kappa)$ for the cofinality of~$\kappa$. A cardinal
 number~$\kappa$ is called regular, if $\kappa =\cof (\kappa)$. We
 define $\co =|\reals|$ and $\omega =|\mathN|$.
 For a set $A\su\reals$ its characteristic function is denoted
 by~$\charf A$. The projection of a set $A\su \reals^2$ onto the
 $x$-axis will be denoted by~$\dom A$. For a cardinal number
 $\kappa\leq\co$ and an open set $U\su\reals^n$ we say that $X\su U$
 is \emph{$\kappa$-dense in~$U$} if $|X\cap V|\geq\kappa$ for
 every nonempty open set $V\su U$.

The symbols $\bfun$ and $\bclass$ will denote the class of all
 bounded functions from $\reals$ to~$\reals$ and the class of all
 families $\A\su \reals ^\reals$ of functions having common
 bound, respectively. For $\F\su\reals^\reals$ we define two
 cardinal invariants
 (cf.\ \cite{TN-RAE} and~\cite{CiesReclaw1}):
\begin{align*}
 \add(\F)& =\min \bigl\{ |\A| \st \A \su \reals^\reals\; \&\;
 \neg (\exists g\in\reals^\reals) (\forall f\in \A)(f+g \in \F)
 \bigr\} \cup \{(2^\co)^+\}\\
 & =\min \bigl\{ |\A| \st \A\su \reals^\reals\; \&\; (\forall
 g\in \reals^\reals) (\exists f\in \A)(f+g \notin \F)\bigr\}
 \cup \{(2^\co)^+\},\\
\intertext{and}
 \addb(\F)& =\min \bigl\{ |\A| \st \A\in \bclass\; \&\; \neg
 (\exists g\in\bfun) (\forall f\in \A)(f+g \in \F) \bigr\}
 \cup \{(2^\co)^+\}\\
 & =\min \bigl\{ |\A| \st \A\in\bclass\; \&\; (\forall g\in\bfun)
 (\exists f\in \A)(f+g \notin \F) \bigr\} \cup \{(2^\co)^+\}.
\end{align*}

The next proposition lists several basic properties of the
 function~$\addb$. The similar properties for the function~$\add$
 can be found in~\cite{CiesReclaw1}. They will be left without
 proofs. (Their easy proofs are analogous to those of
 Propositions~1.1 and~1.3 of~\cite{CiesReclaw1}.)

\prop{prop1}{
 Let $\F \su \reals^\reals$. Then
\begin{description}
 \item[(1)] $\addb(\F) =\addb(\F \cap \bfun)$\textup{;}
 \item[(2)] $\addb(\F) \ge 1$\textup{;}
 \item[(3)] if $\F \su {\mathcal G} \su \reals^\reals$, then
 $\addb(\F) \le \addb({\mathcal G})$\textup{;}
 \item[(4)] $\addb(\F) \le 2$ if and only if
 $\{ f_1 -f_2 \st f_1, f_2 \in \F \cap \bfun \} \neq
 \bfun$\textup{;}
 \item[(5)] $\addb(\F) \ge 2$ if and only if
 $\F \cap \bfun \neq \emptyset$\textup{;}
 \item[(6)] if $\F \sups \bfun$, then $\addb(\F) =(2^\co)^+$.\qed
\end{description}
}

We will find the value of the function $\addb$ for the following
 classes of functions, where $\kappa$~is a cardinal
 number~$\leq\co$ and $X$~is an arbitrary topological space.
\begin{description}
 \item[$\PC{\kappa}$] of all functions $f\colon \reals \to
 \reals$ with the following property: for every $x\in \reals$ and
 every $\e >0$ we have $\bigl| f\cap \bigl[ (x-\e, x) \times
 (f(x) -\e, f(x) +\e) \bigr] \bigr| \ge\kappa$
 and $\bigl| f\cap \bigl[ (x, x+\e)
 \times (f(x) -\e, f(x) +\e) \bigr] \bigr| \ge \kappa$.
 In particular, $\PC{\omega}$~is the class of peripherally
 continuous functions.
 \item[$\Darb$] of all \emph{Darboux functions}
 $f\colon\reals\to\reals$,
 i.e., such that $f[J]$ is an interval for every
 interval $J\su\reals$.
 \item[$\U{\kappa}$] of all functions $f\colon \reals \to \reals$
 fulfilling the following condition: for all $a<b$ and each set
 $A\su [a,b]$ with $|A| <\kappa+1$, the set $f [(a,b) \setminus
 A]$ is dense in the interval
 $\bigl( \min \{ f(a), f(b) \}, \max \{f(a), f(b) \} \bigr)$. In
 particular, $\U{\co}$~is the uniform closure of the
 class~$\Darb$~\cite{BCWeiss-D^u}, while the class~$\U{0}$ was
 examined earlier by T.~Radakovi\v{c}~\cite{Radak} and H.~W.\
 Ellis~\cite{Ellis}. Moreover $\U{\co} =\PC{\co} \cap \U{0}$
 \cite[Theorem~3.2]{BCWeiss-D^u}.
 \item[$\AC$] of all \emph{almost continuous functions} $f\colon
 \reals \to \reals$, i.e., such that for every open set $U\su
 \reals^2$ containing~$f$ there is a continuous function $h\colon
 \reals \to \reals$ with $h\su U$.
 \item[$\Conn(X)$] of all \emph{connectivity functions} $f\colon
 X \to \reals$, i.e., such that the restriction~$f \restr{C}$
 \text{$\bigl($}that is $f\cap \bigl[ C \times \reals
 \bigr]$\text{$\bigr)$} is connected in~$X\times \reals$ whenever
 $C\su X$ is connected.\footnote{Actually we will
 study only the class $\Conn=\Conn(\reals)$, but 
 we need the class $\Conn (\reals \times [0,1])$
 to define the class~$\Ext$.}
 \item[$\Ext$] of all \emph{extendable functions} $f\colon \reals
 \to \reals$, i.e., such that there exists a function $g\in \Conn
 (\reals \times [0,1])$ with $f(x) =g(x,0)$ for
 every~$x\in\reals$.
 \item[$\PR$] of all \emph{functions $f\colon \reals \to \reals$
 with perfect road}, i.e., such that for every $x\in\reals$ there
 exists a perfect set $P\su\reals$ having $x$ as a bilateral
 limit point for which the restriction~$f\restr{P}$ is continuous
 at~$x$.
\end{description}
 For the above classes of functions we have the following proper
 inclusions, marked by arrows.
\begin{center}
\unitlength=1mm
\thicklines
\begin{picture}(130.00,25.00)
\put(3.50,23.00){\makebox(0,0)[ct]{$\Ext$}}
\put(16.00,23.00){\makebox(0,0)[ct]{$\AC$}}
\put(30.00,23.00){\makebox(0,0)[ct]{$\Conn$}}
\put(42.00,23.00){\makebox(0,0)[ct]{$\Darb$}}
\put(51.50,16.00){\makebox(0,0)[ct]{$\U{\co}$}}
\put(65.00,14.00){\makebox(0,0)[cc]{$\cdots$}}
\put(90.00,16.00){\makebox(0,0)[ct]{$\U{\omega}=\cdots =
 \U{0}$}}
\put(3.50,5.00){\makebox(0,0)[ct]{$\PR$}}
\put(51.50,5.00){\makebox(0,0)[ct]{$\PC{\co}$}}
\put(65.00,3.00){\makebox(0,0)[cc]{$\cdots$}}
\put(90.00,5.00){\makebox(0,0)[ct]{$\PC{\omega} \!=\!\cdots
\! =\!\PC{1}$}}
\put(121.50,5.60){\makebox(0,0)[ct]{$\PC{0} \!=\!\reals ^\reals$}}
\put(51.50,11.00){\vector(0,-1){5.00}}
\put(64.50,11.00){\vector(0,-1){5.00}}
\put(78.50,11.00){\vector(0,-1){5.00}}
\put(7.50,21.00){\vector(1,0){5.00}}
\put(19.50,21.00){\vector(1,0){5.00}}
\put(35.00,21.00){\vector(1,0){5.00}}
\put(56.00,14.00){\vector(1,0){5.00}}
\put(68.00,14.00){\vector(1,0){5.00}}
\put(7.50,3.00){\vector(1,0){40.00}}
\put(56.00,3.00){\vector(1,0){5.00}}
\put(68.00,3.00){\vector(1,0){5.00}}
\put(3.50,19.00){\vector(0,-1){13.00}}
\put(44.50,21.00){\vector(2,-3){4.00}}
\put(106.50,3.00){\vector(1,0){5.00}}
\end{picture}
\end{center}
 For the upper row inclusions see, e.g.,~\cite{BHumkeL}. The
 inclusion $\Ext\sun\PR$ follows from~\cite{GibsonRoush-conn-PR}.
 The other relations are more or less evident.

The values of the function~$\add$ for most of these classes have
 been established in several papers, as quoted below.

\begin{theorem}[{\cite{CiesReclaw1}}]   \label{CRmain}
$\add(\PC{\omega}) =2^\co$ and $\add(\Ext) =\add(\PR) =\co^+$.
\end{theorem}

\begin{theorem}[{\cite{CM}}] \label{thCM}
$\co^+ \le \add(\AC) =\add(\Conn) =\add(\Darb) \le 2^\co$,
 $\cof(\add(\Darb)) >\co$, and it is pretty much all that can be
 shown in~ZFC. More precisely, the assertion $\add(\Darb)
 =2^{\co}$ is independent of the cofinality of~$2^{\co}$, and for
 each regular cardinal~$\lambda$ between $\co^+$ and~$2^{\co}$ it
 is consistent with~ZFC that $\add(\Darb) =\lambda$.
\end{theorem}

The values of the function~$\add$ for the other classes have not
 been considered
 so far. However, it is not difficult to find them.
\thm{RestOfA}{
 $\add(\PC{0}) =(2^\co)^+$ and $\add(\U{\kappa})
 =\add(\PC{\kappa}) =2^\co$ for every infinite cardinal
 number~$\kappa\leq\co$.
}

The equality
 $\add(\PC{0}) =(2^\co)^+$ is obvious. The other part
 will be proved in the next section.

It follows from Proposition~\ref{prop1}(6) that
 $\addb(\PC{0})=(2^\co)^+$. It has been
 proved in~\cite{AM-s-b-D} that $\addb(\Darb) =\cof(\co)$. The
 goal of this paper is to establish the values of $\addb$ for all
 other classes by proving the following theorem.
\thm{main}{
\begin{description}
 \item[(1)] $\addb(\Ext) =\addb(\PR) =2$.
 \item[(2)] $\addb(\AC) =\addb(\Conn) =\addb(\Darb)
 =\addb(\U{\co}) =\addb(\PC{\co}) =\cof(\co)$.
 \item[(3)] $\addb(\U{\kappa}) =\addb(\PC{\kappa}) =\co$ for every
 infinite cardinal number~$\kappa<\co$.
\end{description}
}
 In particular, the equality $\addb(\AC)=\cof(\co)$ solves
 Problem~2 of~\cite{GMN}. (This problem was restated
 in~\cite[p.~679]{AM-s-b-D}.) Notice also that Theorem~\ref{main}
 and Proposition~\ref{prop1}(4) imply immediately the following
 corollaries.
\cor{corAC}{
 Every bounded function $f\colon \reals \to \reals$ is the sum of
 two bounded almost continuous functions.\qed
}
\cor{corExt}{
 There exists a bounded function $f\colon \reals \to \reals$
 which is not the sum of two bounded functions with perfect
 road.\qed
}

Notice that Corollary~\ref{corAC} generalizes the theorem of
 U.~B.~Darji
 and P.~D. Humke that every bounded function is the sum of three
 bounded almost continuous functions~\cite{DarjiHumke1}. On the
 other hand, by Corollary~\ref{corExt}, T.~Natkaniec's result
 asserting that every bounded function is the sum of three bounded
 extendable functions~\cite[Theorem~1]{TN-Ext} cannot be improved.

In what follows we will use a generalization of \cite[Lemma~4,
 p.~285]{CPea-i-op} and \cite[Lemma~1]{JC-diff-road}. We will
 need the following notation. For every $\lambda >\omega$ and
 every function $f\colon\reals\to\reals$ let
\begin{align*}
 A^{-}_\lambda(f)& =\bigl\{x\in \reals\st (\exists \e>0)( |f\cap
 [(x-\e,x)\times(f(x)-\e,f(x)+\e)]|<\lambda)\bigr\},\\
 A^{+}_\lambda(f)& =\bigl\{x\in \reals\st (\exists \e>0)( |f\cap
 [(x,x+\e)\times(f(x)-\e,f(x)+\e)]|<\lambda)\bigr\},
\end{align*}
 and $A_\lambda(f)=A^{-}_\lambda(f)\cup A^{+}_\lambda(f)$.
\lem{lemA}{
 If $\lambda$ is a cardinal number with
 $\cof(\lambda)>\omega$, then $|A_\lambda(f)|<\lambda$ for every
 function $f\colon\reals\to\reals$.
}
\begin{proof}
 Assume, by way of contradiction, that
 $|A_\lambda(f)|\geq\lambda$. Suppose that
 $|A^{-}_\lambda(f)| \geq\lambda$, the case
 $|A^{+}_\lambda(f)| \geq\lambda$ being similar. Since
 $\cof(\lambda)>\omega$ and
\[ A^{-}_\lambda(f) =\bigcup _{n\in \mathN} \Bigl\{ x\in\reals \st
 \bigl| f\cap \bigl[ (x-n^{-1},x) \times (f(x) -n^{-1}, f(x)
 +n^{-1}) \bigr] \bigr| <\lambda \Bigr\},
\] there exist an $\e>0$ and a subset~$B$ of~$A^{-}_\lambda(f)$ of
 cardinality~$\geq\lambda$ such that \linebreak  
$\bigl| f\cap \bigl[
 (x-\e,x) \times (f(x)-\e,f(x)+\e) \bigr] \bigr| <\lambda$ for
 every $x\in B$. Choose
 an open interval~$J$ of length~$\e$ such that the set
 $C=\bigl\{ x\in B \st f(x) \in J \bigr\}$ has cardinality
 greater than
 or equal to~$\lambda$. Then, there is an $x_0\in C$ with
 $|(x_0-\e/2,x_0) \cap C| \geq\lambda$. But $f(x_0) \in J$, so
 $J\su (f(x_0)-\e,f(x_0)+\e)$ and $\bigl| f\cap \bigr[
 (x_0-\e,x_0) \times (f(x_0)-\e, f(x_0)+\e) \bigr] \bigr|
 \geq\lambda$, contradicting the fact that $x_0\in B$.
\end{proof}

\section{Proof of Theorem~\ref{RestOfA}}

Let $\kappa \le \co$ be an infinite cardinal. It follows from
 the monotonicity of the function~$\add$ and Theorem~\ref{CRmain}
 that $\add(\U{\co}) \leq \add(\U{\kappa}) \leq \add(\PC{\kappa})
 \leq \add(\PC{\omega})=2^\co$. Thus, it is enough to show that
 $\add(\U{\co}) \geq 2^\co$. This can be established by the
 following minor modification of the proof of
 \cite[Theorem~1.7(3)]{CiesReclaw1}.

Let
 $\F\su\reals^\reals$ be such that $|\F|<2^{\co}$. We will find a
 function $g\colon\reals\to\reals$ such that $f+g\in\U{\co}$ for
 every $f\in \F$.

Let $\G$ be the family of all
 pairs $\la I,J\ra$, where $I$ and~$J$ are nonempty open
 intervals with rational end points. Let
 $\{ \la I_\xi, J_\xi \ra\colon \xi<\co \}$
 be an enumeration of~$\G$ with each pair
 appearing $\co$~many times. For each $\xi<\co$ define a set
 $B_\xi \su I_\xi$ of cardinality~$\co$ such that
 $B_\xi \cap B_\eta =\emptyset$ for any $\xi<\eta<\co$.
 (See, e.g., \cite[Lemma~5]{AM-s-b-D}.)

Next, fix a $\xi<\co$. For each $f\in \F$ construct
 a function $h_{\xi,f} \colon B_\xi\to\rationals$ such that
\[ f(x) +h_{\xi,f}(x) \in J_\xi \quad \textup{for every $x\in B_\xi$.}
\] Then, by \cite[Lemma~2.2]{CiesReclaw1} (used
 with $B=B_\xi$ and $H =\{h_{\xi,f} \st f\in \F\}$), there exists
 a function $g_\xi\colon B_\xi\to\rationals$ such that
 $h_{\xi,f} \cap g_\xi \neq \emptyset$ for every $f\in \F$.

Let $g\colon\reals\to\rationals$ be a common extension of all
 functions~$g_\xi$. Then for every $\xi<\co$ and every $f\in \F$
 there exists an $x\in B_\xi$ such that $f(x)+g(x) \in J_\xi$. So,
 for every $f\in \F$ the graph of $f+g$ is $\co$-dense
 in~$\reals^2$. Thus, $f+g\in\U{\co}$.

\section{Proof of Theorem~\ref{main}(1)}

By Proposition~\ref{prop1}(5) and (3), we have
 $2\leq \addb(\Ext)\leq \addb(\PR)$. Thus, it is enough to prove that
 $\addb(\PR) \leq 2$, i.e., by  Proposition~\ref{prop1}(4), that
 $\{f_1 -f_2\st f_1, f_2 \in \PR\cap\bfun \} \neq \bfun$.
 This follows immediately from the next example.

\ex{ex1}{
 Let $B\su \reals$ be a Bernstein set and $f =\charf{B}
 -\charf{\reals \setminus B}$. If $f =g_0 -g_1$ and $g_0, g_1
 \in \PR$, then both $g_0$ and~$g_1$ are bilaterally unbounded.
}
\begin{proof}
 By way of contradiction assume that there are $g_0, g_1 \in \PR$
 such that $f =g_0 -g_1$ and $g_0$ is not bilaterally unbounded.
 We will assume that $m_0 =\inf g_0 [\reals] >-\infty$. (The
 other case is analogous.) Put $m_1 =\inf g_1 [\reals]$ and
 notice that $m_1 \ge m_0 -1 >-\infty$.

For $i<2$
 choose a nonempty perfect set $P_i$ such that $g_i <m_i +2^{-1}$
 on~$P_i$. Then
\begin{align*}
 &m_0 \le \inf g_0[P_1] = \inf(g_1+f)[P_1] \le (m_1+2^{-1}-1) =
 m_1-2^{-1} <m_1\\
 \intertext{and}
 &m_1 \le \inf g_1[P_0] = \inf(g_0-f)[P_0] \le (m_0+2^{-1}-1) =
 m_0-2^{-1} <m_0,
\end{align*}
 an impossibility. This completes the proof.
\end{proof}

\section{Proof of Theorem~\ref{main}(2)}

By Proposition~\ref{prop1}(3), $\addb(\AC) \leq \addb(\Conn) \leq
 \addb(\Darb) \leq \addb(\U{\co})\leq\addb(\PC{\co})$.
 So, it is enough to prove that $\addb(\PC{\co}) \leq \cof(\co)$
 and $\addb(\AC) \geq \cof(\co)$. The first of these inequalities
 follows from the next example, and the second one follows
 immediately from Theorem~\ref{T1}.
\ex{prop2}{
 There is a family $\A\in\bclass$ with $|\A| =\cof(\co)$ such that
 for every function~$g$ bounded above we have $f+g \notin \PC{\co}$
 for some $f\in \A$.
}
\begin{proof}
 Let $\Aa$ be a family of sets such that $|\Aa| =\cof(\co)$,
 $|A|<\co$ for every $A\in \Aa$, and $\bigl| \bigcup \Aa \bigr|
 =\co$. Clearly we may assume that the sets in~$\Aa$ are nonempty,
 pairwise disjoint, and that $\bigcup \Aa =\reals$. Put $\A =\{
 \charf{A} \st A\in \Aa \}$.

Let $g\colon \reals \to \reals$ be bounded above. Define $M =\sup
 g[\reals]$ and choose an $x\in \reals$ with $g(x) >M-1$. Then
 $x\in A$ for some $A\in \Aa$. Consequently, $(g +\charf{A}) (t)
 =g(t) \le M$ for every $t\notin A$ and $(g +\charf{A}) (x) >M$.
 Thus
\[ 1 \le \bigl| \bigl\{ t\in \reals \st (g +\charf{A}) (t) >M
 \bigr\} \bigr| \le |A| <\co
\] and $g +\charf{A} \notin \PC{\co}$.
\end{proof}

\thm{T1}{
 If $\A \su (-1,1)^\reals$ and $|\A| <\cof(\co)$, then there
 is a function $g\in (-2,2)^\reals$ such that $f+g \in \AC$ for
 every $f\in \A$.
}
\begin{proof}
 For each $f\in\A$ let
\begin{align*}
 U_f& =\bigl\{ \la x, y \ra \st x\in \reals\; \&\; f(x) -2< y
 <f(x) +2 \bigr\} \quad\;\\
 & \textstyle =\bigcup _{x\in\reals} \bigl[ \{x\} \times (f(x)
 -2, f(x) +2) \bigr]
\end{align*}
 and
\[ \K_f =\bigl\{ K\su\reals^2 \st \textup{$K$ is closed \&\
 $|\!\dom (K\cap U_f)| =\co$} \bigr\}.
\] Clearly, $|\K_f|=\co$. Also, by Lemma~\ref{lemA}, $|A_\co(f)|
 <\co$ for every $f\in\A$. So, since $|\A| <\cof(\co)$, we obtain
\begin{equation}\label{con1}
 \textstyle \textup{the set $A =\bigcup _{f\in\A} A_\co(f)$ has
 cardinality less than~$\co$.}
\end{equation}

Let $\bigl\{ \la f_\xi, K_\xi \ra \st \xi <\co \bigr\}$ be a
 transfinite sequence consisting of all pairs $\la f, K \ra$ such
 that $f\in\A$ and $K\in\K_f$. We will construct by transfinite
 induction a sequence $\bigl\{ \la x_\xi, y_\xi \ra \st \xi <\co
 \bigr\}$ such that for every $\xi<\co$ the following conditions
 hold:
\begin{enumerate}
 \item[(i)] $x_\xi \notin \{ x_\zeta \st \zeta<\xi \}\cup A$;
 \item[(ii)] $\la x_\xi, f_\xi(x_\xi) +y_\xi \ra \in K_\xi \cap
 U_{f_\xi}$.
\end{enumerate}

So, assume that $\{ \la x_\zeta, y_\zeta \ra \st \zeta<\xi \}$
 have been already defined for some $\xi<\co$. By the definition
 of~$\K_{f_\xi}$ and condition~\eqref{con1}, we can choose
\[ x_\xi \in \dom (K_\xi \cap U_{f_\xi}) \setminus \bigl( \{
 x_\zeta \st \zeta<\xi \}\cup A \bigr).
\] Let $z_\xi$ be such that $\la x_\xi, z_\xi \ra \in K_\xi \cap
 U_{f_\xi}$ and put $y_\xi =z_\xi -f_\xi(x_\xi)$. Then (i)
 and~(ii) are obviously satisfied. This completes the inductive
 construction.

Notice that $y_\xi \in (-2,2)$ for every
$\xi<\co$ since
$z_\xi \in \bigl( f_\xi(x_\xi)-2, f_\xi(x_\xi)+2 \bigr)$. Define the
 function $g\colon \reals \to (-2,2)$ by
\[ g(x) =\begin{cases}
 y_\xi& \textup{if $x=x_\xi$ for some $\xi <\co$,}\\
 0& \textup{otherwise.}
 \end{cases}
\] 

Fix an $f\in \A$. We will prove that
 $f+g \in \AC$. Let $V\su \reals^2$ be an open set
 containing~$f+g$. First notice that if
\[ E =\dom (U_f \setminus V) =\bigl\{ x\in \reals \st \bigl[ \{x\}
 \times (f(x) -2, f(x) +2) \bigr] \nsu V \bigr\},
\] then $|E| <\co$. Indeed, otherwise $K =\reals^2 \setminus V \in
 \K_f$, so there exists a $\xi<\co$ with $\la f_\xi, K_\xi \ra
 =\la f, K \ra$. By~(ii), we obtain
\[ \la x_\xi, (f+g) (x_\xi) \ra =\la x_\xi, f_\xi(x_\xi) +y_\xi \ra
 \in K_\xi =K,
\] contradicting the fact that $f+g \su V$.

Define
\[ F =\dom \bigl( \bigl[ \reals \times [-1,1] \bigr] \setminus V
 \bigr) =\bigl\{ x\in \reals \st \bigl[ \{x\} \times [-1,1]
 \bigr] \nsu V \bigr\}.
\] Since $[-1,1] \su \bigl( f(x) -2, f(x) +2 \bigr)$ for every
 $x\in\reals$, so $\bigl[ \reals \times [-1,1] \bigr] \su U_f$.
 Hence $F\su E$ and $|F|<\co$. But $\bigl[ \reals \times [-1,1]
 \bigr] \setminus V$ is a closed subset of
 $\reals \times [-1,1]$. Thus $F$~is closed in~$\reals$ and it is
 at most countable.

Let $\J$ be the family of all compact intervals $J =[a,b]$ for
 which there exists a continuous function $h\colon J \to \reals$
 with $h\su V$ and $h(a) =h(b) =0$. Moreover let $G$ be the set
 of all $x\in\reals$ for which there exists a $\delta_x>0$ such
 that $[a,b] \in \J$ whenever $a, b \in (x -\delta_x, x
 +\delta_x) \setminus F$ and $a<b$. The first claim is obvious.

\begin{claim} \label{claim:one}
If $[a_0, a_1] \in \J$ and $[a_1, a_2] 
 \in \J$, then $[a_0, a_2] \in \J$.
\end{claim}

\begin{claim} \label{claim:two} 
If $J =[a, b] \su G$ and $a, b \notin
 F$, then $J\in \J$.
\end{claim}

Indeed, the compactness of~$J$ and the relation $J\su \bigcup
 _{x\in J} (x -\delta _x, x +\delta _x)$ imply that there exist
 $x_0, \dots, x_p \in J$ such that $J\su \bigcup _{i=0} ^p (x_i
 -\delta _{x_i}, x_i +\delta _{x_i})$. Hence we can find
 nonoverlapping
 compact intervals $J_0, \dots, J_l \in \J$ with
 $J =\bigcup _{j=0} ^l J_j$. By Claim~\ref{claim:one}, we obtain $J\in \J$.

\begin{claim} \label{claim:three}
We have $G =\reals$.
\end{claim}

First notice that $G$ is open and that $\reals\setminus F\su G$. By
 way of contradiction suppose that the set $P =\reals\setminus G\su
 F$ is
 nonempty. Thus, $P$ is scattered
 and so, it contains an isolated point~$s$. Let $\delta>0$ be such
 that $[s -\delta, s +\delta] \cap P =\{s\}$. To get a contradiction
 it is enough to show that~$s\in G$.

Let $a, b \in (s -\delta, s +\delta) \setminus F$ and $a<b$. If
 $a>s$ or $b<s$, then, by Claim~\ref{claim:two}, we obtain $[a,b] \in \J$.
 So we may assume
 that $a <s <b$. Let $\e >0$ be such that $a <s -\e <s +\e <b$
 and
\[ R =\bigl[ (s -\e, s +\e) \times ((f+g) (s)-\e, (f+g) (s) +\e)
 \bigr] \su V.
\] (Such an $\e$ exists, since $V$ is open and $\la s, (f+g) (s)
 \ra \in V$.)

Notice that there exist $\la q_0, y_0 \ra, \la q_1, y_1 \ra \in
 U_f \cap R$ such that $q_0 \in (s-\e, s) \setminus E$ and $q_1
 \in (s, s+\e) \setminus E$. We will prove it for $i=0$, the
 other case being similar.

First suppose $s\in A$. Take an arbitrary $q_0 \in (s-\e, s)
 \setminus E$. Then $g(s) =0$, so $(f+g) (s) =f(s) \in (-1,1) \su
 \bigl( f(q_0) -2, f(q_0) +2 \bigr)$. Consequently, the interval
\[ D =\bigl( f(q_0) -2, f(q_0) +2 \bigr) \cap
 \bigl( (f+g)(s) -\e, (f+g)(s) +\e \bigr)
\] is nonempty. So every point $\la q_0, y_0 \ra$ with $y_0 \in D$ has
 the required properties.

Otherwise by the definition of $A_\co^-(f)$, there is a $q_0 \in
 (s-\e, s) \setminus E$ with $f(q_0) \in \bigl( f(s) -\e, f(s)
 +\e \bigr)$. Then
 $|f(s) -f(q_0)|<\e$ and
\[ |(f+g) (s) -f(q_0)| \le |g(s)| +|f(s) -f(q_0)| <2 +\e.
\] Define the set~$D$ as above. Observe that $D\neq \emptyset$ and
 every point $\la q_0, y_0 \ra$ with $y_0 \in D$ has the required
 properties.

Fix an $i<2$. Put $c_i =\min \{ y_i, -1 \}$ and $d_i =\max \{
 y_i, 1 \}$. Since $q_i \notin E$, we see that $\bigl[ \{q_i\}
 \times [c_i, d_i] \bigr] \su \bigl[ \{q_i\} \times (f(q_i) -2,
 f(q_i) +2) \bigr] \su V$. Since the set $\{q_i\} \times [c_i,
 d_i]$ is compact, we can find $a_i, b_i \in (s-\e, s+\e) \setminus
 E$ such that $q_i \in (a_i, b_i)$ and
\[ R_i =\bigl[ (a_i, b_i) \times [c_i,d_i] \bigr] \su V.
\] 

Construct continuous functions $h_0 \colon [a, a_0] \to \reals$
 and $h_1 \colon [b_1, b] \to \reals$ such that $h_0 \cup h_1 \su
 V$ and $h_0 (a) =h_0 (a_0) =h_1 (b_1) =h_1 (b) =0$. (We use
 Claim~\ref{claim:two}.) Extend $h_0 \cup h_1$ to $h\colon [a,b] \to \reals$ by
 connecting the following pairs of points by  straight line
 segments: $\la a_0, 0 \ra$ with $\la q_0, y_0 \ra$,
 $\la q_0, y_0 \ra$ with $\la q_1, y_1 \ra$, and
 $\la q_1, y_1 \ra$ with $\la b_1, 0 \ra$. Notice that these
 segments are contained in rectangles $R_0$, $R$, and $R_1$,
 respectively. Thus $h\su V$. Clearly $h(a) =h(b) =0$ and $h$ is
 continuous, so $[a,b] \in \J$. Hence $s\in G$, an impossibility.
 Claim~\ref{claim:three} has been proved.

Using Claims \ref{claim:one}--\ref{claim:three} it is easy to show that there exists a
 continuous function $h\colon \reals \to \reals$ with $h\su V$.
 So $f+g \in \AC$.
\end{proof} 

\section{Proof of Theorem~\ref{main}(3)}

Let $\kappa <\co$ be an infinite cardinal number. By
 Proposition~\ref{prop1}(3), we have $\addb(\U{\kappa}) \leq
 \addb(\PC{\kappa}) \leq \addb(\PC{\omega})$.
 So, it is enough to prove that $\addb(\PC{\omega}) \leq \co$ and
 $\addb(\U{\kappa}) \geq \co$. The first of these inequalities
 follows from Proposition~\ref{prop3}, and the second one from
 Theorem~\ref{T2}.

\prop{prop3}{
 If a function~$g$ is bounded above, then $g +\charf{\{x\}}
 \notin \PC{\omega}$ for some $x\in \reals$.
}
\begin{proof}
 Define $M =\sup g[\reals]$ and choose an $x\in \reals$ with
 $g(x) >M-1$. Then $(g +\charf{\{x\}}) (t) =g(t) \le M$ for
 $t\neq x$ and $(g +\charf{\{x\}}) (x) >M$. Thus
\[ \bigl| \bigl\{ t\in \reals \st (g +\charf{\{x\}}) (t) >M \bigr\}
 \bigr| =1
\] and $g +\charf{\{x\}} \notin \PC{\omega}$.
\end{proof}

\thm{T2}{
 If $\kappa<\co$, $\A \su (-1,1)^\reals$, and $|\A| <\co$, then
 there is a function
 $g\in (-2,2)^\reals$ such that $f+g \in \U{\kappa}$ for each
 $f\in\A$.
}
\begin{proof}
 The proof is a modification and simplification of
 that of Theorem~\ref{T1}.

Without loss of generality we may assume that $\kappa \ge |\A|
 +\omega$. Let $D$ be a $\kappa$-dense subset of~$\reals$ with
 $|D|=\kappa$, and let
\[ \K=\bigl\{ (p,q)\times\{d\}\st p,q,d\in D\; \&\; p<q \bigr\}.
\] For each $f\in\A$ let
\[ U_f =\bigl\{ \la x, y \ra \st x\in \reals\; \&\; f(x) -2< y <f(x)
 +2 \bigr\}
\] and
\[ \K_f =\bigl\{ K\in\K \st |\!\dom (K\cap U_f)|>\kappa \bigr\}.\qquad
 \qquad \;\quad
\] Clearly, $|\K_f|=\kappa$. Also, by Lemma~\ref{lemA},
\[ \textstyle \textup{the set $A=\bigcup_{f\in \A} A_{\kappa^+}(f)$
 has cardinality $\leq\kappa$.}
\] 

Let
 $\{\la f_\xi, K_\xi \ra \st \xi <\kappa \}$ be a transfinite
 sequence consisting of all pairs $\la f, K \ra$ such that
 $f\in\A$ and $K\in\K_f$. As in Theorem~\ref{T1} we can construct
 by transfinite induction a sequence $\{\la x_\xi, y_\xi \ra \st
 \xi <\kappa\}$ such that for every $\xi<\kappa$
 the following conditions hold:
\begin{enumerate}
 \item[(i)] $x_\xi \notin \{ x_\zeta \st \zeta<\xi \}\cup A$;
 \item[(ii)] $\la x_\xi, f_\xi(x_\xi) +y_\xi \ra \in K_\xi\cap
 U_{f_\xi}$.
\end{enumerate}
 Notice that $y_\xi \in (-2,2)$ for every $\xi<\kappa$. Define
 $g\colon \reals \to (-2,2)$ by
\[ g(x) =\begin{cases}
 y_\xi& \textup{if $x=x_\xi$ for some $\xi <\kappa$,}\\
 0& \textup{otherwise.}
 \end{cases}
\] 

Fix an $f\in \A$. We will prove that $f+g \in \U{\kappa}$. Let
 $a<b$ and define $c=(f+g)(a)$ and $d=(f+g)(b)$. Clearly we may
 assume that $c\neq d$. We will assume that $c<d$, the case $d<c$
 being similar.

Let $y\in(c,d)\cap D$. We will find $p,q \in (a,b) \cap D$
 such that $p<q$ and
\begin{equation}\label{con33}
 K =(p,q) \times \{y\} \in \K_f.
\end{equation}

Indeed, if $y\in(-1,1)$, then $\bigl[ \reals \times \{y\} \bigr]
 \su U_f$ and the relation~\eqref{con33} holds for any $p,q \in
 (a,b) \cap D$ with $p<q$. So, suppose that $y\notin (-1,1)$. We
 will assume that $y\geq 1$, the case $y\leq -1$ being essentially
 the same.

Then $f(b) +g(b) =d >y \geq 1 >f(b)$, so $g(b)\neq 0$. In
 particular, $b\notin A_f^-$. So, there exist an $\e\in(0,d-y)$ and
 $p,q \in (a,b) \cap D$ such that the set $S= \break 
\{ x\in (p,q) \st f(x)
 >f(b) -\e\}$ has cardinality greater than~$\kappa$. But for every
 $x\in S$ we have
\[ f(x)+2 >f(x)+g(b) >f(b)-\e +g(b) =d-\e >y \geq 1 >f(x)-2.
\] Thus $S\times \{y\} \su U_f$ and $(p,q)\times \{y\} \in \K_f$.

Let $\xi<\kappa$ be such that $\la f,K\ra=\la f_\xi,K_\xi\ra$. Then
 $x_\xi\in (p,q) \su (a,b)$ and $(f+g)(x_\xi)=f_\xi(x_\xi)+y_\xi=y$.
 It follows that $(c,d)\cap D \su (f+g) [(a,b)]$, i.e., that $(f+g)
 [(a,b)]$ is $\kappa$-dense in $(c,d)$. This completes the proof.
\end{proof}

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