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\markright{Cardinals ...
\ \ \today }


\title{Covering Baire 1 functions with Darboux functions and the
cofinality  of the ideal of meager sets}

%\MathReviews{Primary:  26A15; Secondary: 03E75, 54A25.}
%\keywords{cardinal functions; extendable, Darboux, almost continuous
%and peripherially continuous functions; functions with perfect road. }


\author{{\small Francis Jordan}%
\thanks{AMS classification numbers: Primary 26A15;  Secondary 54A25
\endgraf  Key words and phrases: cardinal functions, meager sets, Baire
class 1,  extendable functions, connectivity functions, peripherially
continuous functions, almost continuous functions, Darboux functions.
\endgraf  This paper was written under supervision of K.~Ciesielski. The
author wishes to thank him for many helpful conversations.},
\small  Department of Mathematics, West Virginia University,\\
Morgantown, WV 26506-6310}

\date{}


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\newcommand{\ext}{{\operatorname {Ext}}}
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\newcommand{\phc}{{\operatorname {Pc}}}
\newcommand{\swiat}{{\operatorname {Sw}}}
\newcommand{\quasi}{{\operatorname {Qc}}}
\newcommand{\cont}{{\operatorname {cont}}}
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%\title{The minimum number of Darboux functions needed to cover Baire
%class 1}
%\author{Francis Jordan\\
%\small Dept. of Mathematics, West Virginia Univ., Morgantown, WV
%26506-6310}

\date{}


\pagestyle{myheadings}

%\markright{ ...
%\ \ \today }

\begin{document}\maketitle

\begin{abstract}
Let $\dar$ stand for the Darboux Baire class 1 functions.
We show that the cofinality of the meager sets in $\real$ is the smallest
cardinality of a set of Baire class 1 functions $F$ such that for any
finite collection of Baire class 1 functions $G$ there is an $f\in F$ such
that $G\subseteq f+D$.  Other results of this
type are shown.  These results are then considered as statements about
additivity.  The notion of super-additivity is introduced.
\end{abstract}
\section{Preliminaries} We use standard notation as in \cite{Ci:book}.
In particular,  for a set $X$ we denote its cardinality  by $|X|$.  Given
sets $X$ and $Y$ we denote by $Y^X$ the set of all  functions from $X$
into $Y$.  If $\kappa$ is a cardinal number and $X$ is a  set we let
$[X]^{<\kappa}$ denote the collection of subsets of $X$ which have
cardinality strictly less than $\kappa$.  We let $\real$ denote the set
of real  numbers and $\rational$ stand for the set of rational numbers.
The cardinalities of $\real$ and $\rational$ will be denoted by $\cuum$
and
$\omega$, respectively.  If $A\subseteq\real$  we let $\charf{A}$ stand
for the characteristic function of $A$.   The closure, boundary, and
interior of a set $X\subseteq\real$ will be written,  respectively, as
$\cl(X)$, $\bd(X)$, and $\interior(X)$.   Let $\cnwd$, $\meager$, and
$\dense$ denote  the closed nowhere dense, meager, and co-meager subsets
of
$\real$, respectively.   For a function $f\colon\real\to\real$ we let
$\cont(f)$ denote the set of points at which $f$ is continuous.   For a
function $f\colon\real\to\real$ and a point $x\in\real$ we define the
oscillation of $f$ at $x$ to be
\begin{equation*}
\osc(f,x)=\lim_{\delta\to 0^+}\sup\{|f(x)-f(t)|\colon t\in
(x-\delta,t+\delta)\}.
\end{equation*}
Note that $f$ is continuous at $x$ if and only if
$\osc(f,x)=0$. For $n\in\omega$ we let $\osc_n(f)$ denote the set
$\{x\in\real\colon\osc(f,x)\geq 1/2^n\}$.  The oscillation function is
upper semi-continuous so we have the following fact,

\prop{prop:osc}{$\cl(osc_n(f))=\osc_n(f)\subseteq\osc_{n+1}(f)$ for any
$f\in\real^{\real}$ and $n\in\omega$.}
The oscillation of a function $f\colon\real\to\real$ on a set $S$ is defined to
be $\osc(f,S)=\sup\{|f(x)-f(y)|:x,y\in S\}$.


Given $f\in\real^{\real}$, $x\in\real$, and $S\subseteq\real$ we  define
the right cluster set of
$f$ with respect to $S$ at $x$ by
\begin{equation*}
\wacc^{+}_{S}(f,x)=\bigcap_{n=1}^{\infty}(\cl(f[(x,x+1/n)\cap S])).
\end{equation*} We define the left cluster set of $f$ with respect to $S$
at $x$, denoted   by $\wacc^{-}_{S}(f,x)$, in a similar way.  Finally,
we let
$\wacc_{S}(f,x)=\wacc_{S}^{-}(f,x)\cap\wacc_{S}^{+}(f,x)$ denote  the
bilateral cluster set of $f$ with respect to $S$ at $x$.   We define the
strong right cluster set of $f$  with respect to $S$ at $x$ by
\begin{equation*}
\acc^{+}_{S}(f,x)=\bigcap_{n=1}^{\infty}(f[(x,x+1/n)\cap S]).
\end{equation*} The strong left cluster set of $f$ with respect to $S$ at
$x$ is defined in a similar manner and denoted by
$\acc^{-}_{S}(f,x)$.  We call $\acc_{S}^{-}(f,x)\cap\acc_{S}^{+}(f,x)$
the  strong bilateral cluster set of $f$ with respect to $S$ at
$x\in\real$  and denote it by $\acc_{S}(f,x)$.   If
$f\colon\real\to\real$ is a function  we let $\norm(f)=\sup\{|f(x)|\colon
x\in\real\}$.

If a function $f\colon\real\to\real$ is the  pointwise limit of a
sequence of continuous functions we say
$f$ is a Baire class one function.  Let $\baire_1$ denote  the Baire
class one functions.    We say $f\colon\real\to\real$ is a cliquish
function provided that for every $x_0\in\real$ and $\epsilon>0$ and
neighborhood
$W$ of $x_0$ there is a nonempty open set $W_0\subseteq W$ such that
$\osc(f,{W_0})<\epsilon$.   We
denote the family of cliquish functions by $\cliq$.  It is well known,
and easy to prove,
that if $f\in\cliq$ then $\cont(f)$ is a co-meager subset of $\real$.  We
note the
following facts about Baire class one and cliquish  functions.
\prop{prop:aaa}{
\begin{description}
\item[(1)] $\baire_1\subseteq\cliq$,
\item[(2)] $\osc_{n}(f)$ is closed and nowhere dense for any
$f\in\cliq$ and $n\in\omega$, and
\item[(3)] $\baire_1$ and $\cliq$ are additive groups.
\end{description}}
\proof The containment $\baire_1\subseteq\cliq$ follows from
\cite[p.394]{KUR}.  Item (2) follows immediately from Proposition~\ref{prop:osc}.
The cliquish functions form an
additive group since, if $f,g\in\cliq$ then
$\cont(f+g)\supseteq\cont(f)\cap\cont(g)$  and
$\cont(f)\cap\cont(g)$ must be a co-meager set by the Baire Category
Theorem.  The Baire class one functions form an  additive group because
the sum of the limits of  two pointwise convergent sequences of functions
is equal to the  pointwise limit of the sums of the terms of the
sequences.\qed


\section{Introduction} We will be concerned with some cardinal invariants
of the real line  related to the  algebraic properties of the following
families of functions in
$\baire_1$ and $\cliq$.  We give descriptions of these families for
general topological spaces, although our discussion will be  restricted
to the real line.  To find out more about  the families below see
\cite{GNsurvey}, \cite{NATsurvey}, and
\cite{KCsurvey}.
\begin{description}
\item[$\dar$:] $f\in Y^X$ is a {\em Darboux} function if and only if
$f[C]$ is connected  in $Y$ for every connected subset $C$ of $X$.
\item[$\conn$:] $f\in Y^X$ is a {\em connectivity} function if and only
if the graph of
$f$ restricted to $C$ is connected in $X\times Y$ for every connected
subset $C$ of $X$.
\item[$\acon$: ] $f\in Y^X$ is an {\em almost continuous} function  if
and only if every open set in $X\times Y$ containing $f$ also  contains
some continuous function $g\in Y^X$.
\item[$\ext$:] $f\in Y^X$ is an {\em extendable} function if and only if
there is a connectivity function $g:X\times [0,1]\to Y$ such that
$f(x)=g(0,x)$ for every $x\in X$.
\item[$\pr$: ] $f\in\real^{\real}$ is a function with a {\em perfect
road}  if and only if for every $x\in\real$ there is a perfect set $P$
such that $x$ is a bilateral limit point of $P$ and the restriction
$f|_P$ is continuous at $x$.
\item[$\phc$: ] $f\in Y^X$ is a {\em peripherally continuous} function
if and only if  for every $x\in X$ and every pair of open sets $U\subset X$ and
$V\subset Y$  such that $x\in U$ and $f(x)\in V$ there is an  open
neighborhood $W$ of $x$ with $\cl(W)\subset U$ and
$f[\bd(W)]\subseteq V$, where $\bd(W)$ denotes the boundary of $W$.
\item[$\quasi$:] $f\in Y^X$ is a {\em quasi-continuous} function  if and
only if  at each point $p\in X$ the following condition holds:  for every
open set $U\subseteq X$ with
$p\in U$ and open set $V\subseteq Y$ with $f(p)\in V$ there exists  a
non-empty open set $W\subseteq U$ such that $f[W]\subseteq V$.
%\item[$\wsw$:] $f\in \real^{\real}$ is a {\'S}wi{\c a}tkowski function
%if %and
%only if for every $a<b$ there exists $y$ between $f(a)$ and $f(b)$ such
%that there is an
%$x\in\cont(f)\cap (a,b)$ such that $f(x)=y$.
\item[$\swiat$:] $f\in \real^{\real}$ is a {\em strong {\'S}wi{\c
a}tkowski} function if and only if for every $a<b$ and $y$ strictly
between $f(a)$ and $f(b)$ there is an
$x\in\cont(f)\cap (a,b)$ such that $f(x)=y$.
\end{description}

We collect some facts about the above classes which will be of  use in
what follows.
\prop{prop:2}{The following containments and equalities hold and all
containments mentioned are proper.
\begin{description}
\item[(i)] If $\F\in\{\ext,\acon,\pr,\conn,\phc\}$ then
$\dar\cap\baire_1=\F\cap\baire_1$,
\item[(ii)] $\swiat\subseteq\dar\cap\quasi$,
\item[(iii)] $\quasi\subseteq\cliq$,
\item[(iv)] $\quasi\cap\pr=\quasi\cap\phc$, and
\item[(v)] $\pr\cup\dar\subseteq\phc$.
\end{description}}
\proof The containment of (ii) is straightforward and left without
proof.  The following
example \cite[p.10]{AM} shows that the containment  of (ii) is proper
\begin{equation*} g(x)=
\begin{cases} 1+x+\text{sin}(1/x)& \text{if $x>0$;}\\
-1+x+\text{sin}(1/x)& \text{if $x<0$}\\ 0& \text{if $x=0$.}
\end{cases}
\end{equation*}
It is clear from the definitions that $\quasi\subseteq\cliq$, the
characteristic
function of a point show that the containment is proper.  Thus, (iii) holds.  We
give the citations for (i).  The equalities
$\dar\cap\baire_1=\F\cap\baire_1$  for $\F=\ext,\acon,\pr,\conn,\phc$ are
shown in \cite{BHL},
\cite{JB}, \cite{MAX}, \cite{KKWS}, and \cite{JY} respectively.   The
equality $\quasi\cap\pr=\quasi\cap\phc$ is shown in \cite{GR}.  For  (v)
see \cite{BHL} and \cite{GR}.\qed

In \cite[p.17]{AM} A.~Maliszewski proved.
\prop{prop:4}{Let $H\in[\cliq]^{<\omega}$.   There
exists an $f\in\swiat\cap\baire_1$ such that
$f+H\subseteq\swiat$ and $\bigcap\{\cont(g)\colon g \in H\}\subseteq\cont(f)$.
Moreover, given $\epsilon>0$ we may assume that
$\norm(f)\leq\sup\{\osc(g,x)\colon g\in H \&\ x\in\real\}+\epsilon$.}

It was also shown in \cite[p.19]{AM} that the above proposition could
not be improved to include infinite families of functions.
\prop{prop:45}{For any function $g$, if $\cont(g)\neq\emptyset$ then
there is a $q\in\rational$ such that
$g+\charf{\{q\}}\notin\dar\cup\quasi$.}

The two propositions above may interpreted as statements about a cardinal
function which has been studied for families of real functions in  more
general settings \cite{KCsurvey}.   Given a family
$\F\subseteq\real^{\real}$ the additivity of
$\F$, denoted by $\add(\F)$, is defined to be
\begin{equation*}
\add(\F)=\min\left(\{|F|\colon F\subseteq \real^{\real}\ \&\
\left(\forall   g\in \real^{\real}\right) (\exists f\in F) (f+g\notin
\F)\}\cup \{(2^{\cuum})^{+}\}\right).
\end{equation*} The above definition may be restricted to smaller classes
of  functions in the following way.  Let $\cal{H}\subseteq \real^{\real}$
and $\F\subseteq\real^{\real}$ then the additivity of $\F$ in
${\cal H}$,  denoted by $\add_{{\cal H}}(\F)$, is defined to be
\begin{equation*}
\add_{{\cal H}}(\F)=\min\left(\{|F|\colon F\subseteq {\cal H}\ \&\
\left(\forall   g\in{\cal H}\right) (\exists f\in F) (f+g\notin\F)\}\cup
\{|{\cal H}|^{+}\}\right).
\end{equation*} The following proposition is an adaptation of
\cite[Proposition 1]{jord1} which dealt with the special case
${\cal H}=\real^{\real}$.


\noindent\prop{prop:667}{Let ${\mathcal P},\F,{\mathcal H}\subseteq
\real^{\real}$
and
$\charf{\emptyset}\in{\mathcal H}$.  Then,
\begin{description}
\item[(i)] if $\F\cap{\mathcal H}=\emptyset$, then $\add_{{\mathcal H}}(\F)=1$;
\item[(ii)]  if $\F\cap{\mathcal H}={\mathcal H}$, then $\add_{{\mathcal
H}}(\F)=|{\mathcal H}|^+$;
\item[(iii)] if $\F\subseteq{\mathcal P}$, then $\add_{{\mathcal
H}}(\F)\leq\add_{{\mathcal H}}({\mathcal P})$; and
\item[(iv)] if ${\mathcal H}$ is an additive group with more than  one
element and $\F\cap {\mathcal H}\neq\emptyset$, then $2=\add_{{\mathcal H}}(\F)
\text{ if and only if } (\F\cap{\mathcal H}) - (\F\cap{\mathcal
H})\neq{\mathcal H}$;
\item[(v)] if $2<\add_{{\mathcal H}}(\F)
\text{, then } (\F-\F)\cap{\mathcal H}={\mathcal H}$.
\end{description}}
\proof We show (i).  If $\F\cap{\mathcal H}=\emptyset$, then
$\charf{\emptyset}$ has the property that
$\charf{\emptyset}+h\notin\F$ for all $h\in {\mathcal H}$.   Thus,
$\add_{{\mathcal H}}(\F)=1$.

We show (ii).  Suppose $\add_{{\mathcal H}}(\F)<|{\mathcal H}|^+$.  Then there
is an $F\subseteq {\mathcal H}$ with the property that for every $h\in {\mathcal
H}$ there is  some $f\in F$ such that $f+h\notin\F$.  In particular,
there is some
$f\in F\subseteq H$ such that $f=f+\charf{\emptyset}\notin\F$.

Since the proof of (iii) differs only slightly from the proof of its
equivalent statement in \cite[Proposition 1]{jord1}, we
exclude the proof.

We show (iv).  Suppose that $(\F\cap{\mathcal H}) - (\F\cap{\mathcal
H})={\mathcal
H}$.   We consider the two possible cases.  Let
$S=\{|F|\colon F\subseteq {\mathcal H}\ \&\ (\forall g\in{\mathcal H})  (\exists
f\in F) (f+g\notin\F)\}$.  First assume $S=\emptyset$.   Then
$\add_{{\mathcal H}}(\F)=|{\mathcal H}|^+>2$.  Now assume that there exists an
$F\in S$.  We show that $|F|>2$.   By way of contradiction, assume that
$F=\{f_1,f_2\}\in[{\mathcal H}]^{<3}$.   Since $(\F\cap{\mathcal H}) -
(\F\cap{\mathcal H})={\mathcal H}$, there exist
$h_1,h_2\in\F\cap{\mathcal H}$ such that $f_1-f_2=h_1-h_2$.  Let
$g=f_1-h_1=f_2-h_2$.  Notice that $g\in{\mathcal H}$ since ${\mathcal H}$ is a
group.   For $i\in\{1,2\}$, we have $f_i+g=f_i+(h_i-f_i)=h_i\in\F$.  Thus,
there is  a $g\in{\mathcal H}$ such that $g+F\subseteq\F$, which contradicts
our choice of
$F$.  Hence, $\add_{{\mathcal H}}(\F)>2$.  To see the other implication
suppose that $(\F\cap{\mathcal H}) - (\F\cap{\mathcal H})\neq{\mathcal H}$.
Since
${\mathcal H}$  is a group and $\F\cap{\mathcal H}\neq\emptyset$ it is easy to
see that
$\add_{\mathcal H}(\F)\geq 2$.  So we show that $\add_{\mathcal H}(\F)\leq 2$.
Pick
$h\in{\mathcal H}\setminus((\F\cap{\mathcal H}) - (\F\cap{\mathcal H}))$
and put
$F=\{h,\charf{\emptyset}\}$.  Let $g\in{\mathcal H}$ be arbitrary.  It is
enough to  show that $f+g\notin\F$ for some $f\in F$.  However, if
$g=\charf{\emptyset}+g\in\F$ and $h+g\in\F$ then, since ${\mathcal H}$ is a
group,  we have $h\in(F\cap{\mathcal H})-g\subseteq (\F\cap{\mathcal H}) -
(\F\cap{\mathcal H})$,  contradicting the choice of $h$.

We prove (v). Suppose that $(\F-\F)\cap{\mathcal H}\neq{\mathcal H}$.  Pick
$h\in {\mathcal H}\setminus(\F-\F)$ and let
$F=\{\charf{\emptyset},h\}$.  By way of contradiction, assume there  is
some $g\in{\mathcal H}$ such that $g+F\subseteq\F$.  Then,
$g=g+\charf{\emptyset}\in\F$, so
$h\in\F-g\subseteq (\F-\F)$ which contradicts the choice of
$h$.  Thus, $\add_{{\mathcal H}}(\F)\leq 2$.\qed

Additivity and its restricted versions have a nice
interpretation in terms of coverings of one family of functions by
another.  We state this relationship in the following proposition:

\noindent\prop{prop:47}{Let $\F,{\mathcal H}\subseteq\real^{\real}$ and
$\add_{{\mathcal H}}(\F)<|{\mathcal H}|^+$.   Then $\add_{{\mathcal
H}}(\F)$ is the
minimum cardinality of a set
$F\subseteq{\mathcal H}$ such that ${\mathcal H}\cap \bigcap\{-f+\F\colon f\in
F\}=\emptyset$; or, equivalently, that ${\mathcal
H}\subseteq\bigcup\{-f+(\real^{\real}\setminus\F)\colon  f\in F\}$.}
\proof  Since $\add_{{\mathcal H}}(\F)<|{\mathcal H}|^+$, there is an
$F\subseteq{\mathcal H}$ that witnesses the main part of the definition of
$\add_{\mathcal H}(\F)$, i.e., $|F|=\add_{{\mathcal H}}(\F)$ and
\begin{equation}\label{eq3:wart} (\forall h\in{\mathcal H}) (\exists f\in F)
(h+f\notin\F).
\end{equation} We claim that ${\mathcal H}\cap\bigcap\{-f+\F\colon f\in
F\}=\emptyset$.   To see this, assume there is some
$h\in{\mathcal H}\cap\bigcap\{-f+\F\colon f\in F\}$.   Such an $h$ would
have the property that $h+f\in\F$ for each
$f\in F$.  But this would  contradict (\ref{eq3:wart}) since $h\in{\mathcal
H}$.   Thus, \[{\mathcal H}\cap\bigcap\{-f+\F\colon f\in F\}=\emptyset\]  for
some $F$ of cardinality $\add_{{\mathcal H}}(\F)$.

Now assume $F\subseteq{\mathcal H}$ and $|F|<\add_{{\mathcal H}}(\F)$.
There is
an $h\in{\mathcal H}$ such that $h+F\subseteq\F$.   So, $h\in -f+\F$ for each
$f\in F$.  Thus,
${\mathcal H}\cap\bigcap\{-f+\F\colon g\in F\}\neq\emptyset$ for any
$F$ such that $|F|<\add_{{\mathcal H}}(\F)$, which completes the  proof.\qed

Using the language of additivity, we will now state some corollaries  to
Propositions \ref{prop:4} and \ref{prop:45}.
\noindent\cor{cor3:1}{If
$\F\in\{\swiat,\dar\cap\quasi,\dar,\quasi,\dar\cup\quasi\}$, then
\[\add_{\cliq}(\F)=\add_{\baire_1}(\F)=\omega.\]}
\proof By containment, we have that
$\add_{\cliq}(\swiat)\leq
\add_{\cliq}(\F)\leq\add_{\cliq}(\quasi\cup\dar)$ for
$\F\in\{\dar,\quasi,\swiat,\dar\cap\quasi,\dar\cup\quasi\}$.  Since
$\baire_1\subseteq\cliq$, it follows from Proposition~\ref{prop:4}  that
$\omega\leq\add_{\cliq}(\swiat)$.  Using  Proposition~\ref{prop:45}, the
fact that any cliquish function must have at least one point of continuity, and
noting that the  characteristic function of a point is cliquish, we have
$\add_{\cliq}(\dar\cup\quasi)\leq\omega$.  It now follows that
$\omega=\add_{\cliq}(\F)$ for all $\F\in\{\dar,\quasi,\swiat,
\dar\cap\quasi,\dar\cup\quasi\}$.   A similar argument yields the
equalities for $\add_{\baire_1}$.\qed

Corollary~\ref{cor3:1} together with Proposition~\ref{prop:47} implies
that the minimum number of translations of $\baire_1\setminus\dar$ by
Baire class one functions required to cover Baire class one is $\omega$.

To get a statement similar to that of  Corollary~\ref{cor3:1} for
quasi-continuous functions, we need to make a minor modification of
Proposition~\ref{prop:45}.  In particular, the characteristic  functions
of singletons are not quasi-continuous so  Proposition~\ref{prop:45} will
not be useful to us  in the quasi-continuous case.  So, we will consider
characteristic functions of open intervals, which are quasi-continuous.
It should be pointed out that the proof here is  essentially identical to
the one which appears in \cite{AM} for  Proposition~\ref{prop:45}.
\noindent\prop{prop:46}{For any function $g$, if $\cont(g)\neq\emptyset$, then
there is a $q\in\rational$ and an $n\in\omega$ such that
$g+\charf{(q-1/n,q+1/n)}\notin\dar$.}
\proof Suppose $g$ is a function such that $\cont(g)\neq\emptyset$.   Let
$x\in\cont(g)$.  There is a $\delta>0$ such that
$g[(x-\delta,x+\delta)]\subseteq (g(x)-1/3,g(x)+1/3)$.   Pick
$q\in\rational$ and $n\in\omega$ such that
$[q-1/n,q+1/n]\subseteq (x-\delta,x+\delta)$.   For
$w\in (q-1/n,q+1/n)$, we have
\[(g+\charf{(q-1/n,q+1/n)})(w)>1+g(x)-1/3=g(x)+2/3.\]   For points
$z\in (x-\delta,x+\delta)\setminus (q-1/n,q+1/n)$, we have
\[(g+\charf{(q-1/n,q+1/n)})(z)<g(x)+1/3.\]  Thus,
$(g+\charf{(q-1/n,q+1/n)})[(x-\delta,x+\delta)]$ is not an interval.
Therefore, we must conclude that $g+\charf{(q-1/n,q+1/n)}$ is not Darboux.\qed

\noindent\cor{cor3:2}{$\add_{\baire_1\cap\quasi}(\swiat)=
\add_{\baire_1\cap\quasi}(\dar)=
\add_{\quasi}(\dar)=\add_{\quasi}(\swiat)=\omega$.}
\proof We first show that
$\add_{\quasi}(\swiat)=\add_{\quasi}(\dar)=\omega$.   Since
$\swiat\subseteq\quasi\subseteq\cliq$, it follows from
Proposition~\ref{prop:4} that $\omega\leq\add_{\quasi}(\swiat)$.   The
countable family of functions constructed in  Proposition~\ref{prop:46}
were all quasi-continuous and any quasi-continuous function has a point
of continuity; so, we have
$\add_{\quasi}(\dar)\leq\omega$.  Finally, we have
$\add_{\quasi}(\swiat)\leq\add_{\quasi}(\dar)$.   Thus,
$\add_{\quasi}(\swiat)=\add_{\quasi}(\dar)=\omega$.   A similar argument
will give the equalities for
$\add_{\quasi\cap\baire_1}$.\qed

Corollary~\ref{cor3:1} tells us that for any finite collection $G$  of
Baire one functions there is an $f\in\baire_1$ such that
$f+G\subseteq\dar$.  One may ask what the minimal cardinality of a
family of Baire one functions $F$ such that for every finite family of
functions $G$ there is an $f\in F$ such that $f+G\subseteq\dar$ is.   This
question leads us to define a new cardinal function which,  like
additivity, has meaning in more general settings \cite{jord4} and
also has a nice interpretation in terms of coverings.  If
${\mathcal H},\F\subseteq\real^{\real}$,  then
the {\it super-additivity} of $\F$ in ${\mathcal H}$ is defined to be
\begin{equation*}
\add^{*}_{{\mathcal H}}(\F)=\min\left\{|F|\colon F\subseteq{\mathcal H}
\ \&\ \left(\forall G\in [{\mathcal H}]^{<\add_{{\mathcal H}}(\F)}\right)
(\exists f\in F) (f+G\subseteq\F)\right\}
\end{equation*}
This notion of super-additivity was developed by myself and T.~Natkaniec
during his visit to West Virginia University in October of 1997.

We now state and prove some basic facts about super-additivity.
\noindent\prop{prop:sup}{Let $\F,{\mathcal E},{\mathcal
H}\subseteq\real^{\real}$ and
$\charf{\emptyset}\in {\mathcal H}$.  Then
\begin{description}
\item[(i)] if $\F\cap{\mathcal H}={\mathcal H}$, then $\add^*_{{\mathcal
H}}(\F)=1$;
\item[(ii)] if $\F\cap{\mathcal H}=\emptyset$, then
$\add_{{\mathcal H}}^*(\F)=1$;
\item[(iii)] if $\add_{{\mathcal H}}(\F)=\add_{{\mathcal H}}({\mathcal E})$ and
$\F\subseteq{\mathcal E}$ then,
$\add^*_{{\mathcal H}}(\F)\geq\add^*_{{\mathcal H}}({\mathcal E})$.
\end{description}}
\proof We show (i).  If ${\mathcal H}=\F\cap{\mathcal H}$ then, by
Proposition~\ref{prop:667}(ii),
$\add_{{\mathcal H}}(\F)=|{\mathcal H}|^+$.  Let
$G\in[{\mathcal H}]^{<|{\mathcal H}|^+}$.  Clearly,
$\charf{\emptyset}+G\subseteq{\mathcal H}=\F\cap{\mathcal H}$.  So,
$\add_{{\mathcal H}}^*(\F)=1$.

We show (ii).  If $\F\cap{\mathcal H}=\emptyset$ then, by
Proposition~\ref{prop:667}(i),
$\add_{{\mathcal H}}(\F)=1$.  Since $[{\mathcal H}]^{<1}=\{\emptyset\}$ and
$\charf{\emptyset}+\emptyset\subseteq \F$, it follows that
$\add_{{\mathcal H}}^*(\F)=1$.

We show (iii).  Let $\kappa=
\add_{{\mathcal H}}(\F)=\add_{{\mathcal H}}({\mathcal E})$.  Suppose
$F\subseteq{\mathcal H}$ and $|F|<\add^*_{{\mathcal H}}({\mathcal E})$.
Then, there
exists a $G\in[{\mathcal H}]^{<\kappa}$ such that
$f+G$ is not contained in ${\mathcal E}$ for every $f\in F$.   But
$\F\subseteq{\mathcal E}$; so, $f+G$ is not contained in
$\F$ for every $f\in F$.  Thus,
$\add^*_{{\mathcal H}}(\F)\geq\add^*_{{\mathcal H}}({\mathcal E})$.\qed

The next proposition brings out the covering interpretation  of this
cardinal function.
\noindent\prop{prop:52}{Let $\charf{\emptyset}\in{\mathcal
H}\subseteq\real^{\real}$
and
$\F\subseteq\real^{\real}$ be such that ${\mathcal H}\cap\F\neq{\mathcal H}$.
Then, $\add_{{\mathcal H}}^*(\F)$ is the  minimum cardinality of a family
$F\subseteq{\mathcal H}$  such that
\begin{equation}\label{eq3:cov}
[{\mathcal H}]^{<\add_{{\mathcal H}}(\F)}\subseteq
\bigcup_{f\in F}[-f+\F]^{<\add_{{\mathcal H}}(\F)}.
\end{equation}}
\proof Let $F\subseteq{\mathcal H}$ be such that
$|F|=\add_{{\mathcal H}}^*(\F)$ and
\begin{equation*}
\left(\forall   G\in [{\mathcal H}]^{<\add_{{\mathcal H}}(\F)}\right) (\exists
f\in F)  (f+G\subseteq\F).
\end{equation*} We show $F$ satisfies (\ref{eq3:cov}).  Suppose
$G\in[{\mathcal H}]^{<\add_{{\mathcal H}}(\F)}$.  There is an $f\in F$ such
that
$f+G\subseteq\F$.  This means that
$G\subseteq-f+\F$.  Thus,
$G\in [-f+\F]^{<\add_{{\mathcal H}}(\F)}$.   So, $F$ satisfies (\ref{eq3:cov}).

Suppose now that $F\subseteq{\mathcal H}$ satisfies (\ref{eq3:cov}).   Let
$G\in [{\mathcal H}]^{<\add_{\mathcal H}(\F)}$.  By (\ref{eq3:cov}),  there
is an
$f\in F$ such that
$G\in [-f+\F]^{<\add_{{\mathcal H}}(\F)}$.  So,
$f+G\subseteq\F$.  Thus, $|F|\geq\add^*_{{\mathcal H}}(\F)$.\qed

A basic relationship between additivity and
super-additivity observed by myself and T.~Natkaniec is stated
in the next proposition.
\noindent\prop{prop:51}{If $\F,{\mathcal H}\subseteq\real^{\real}$ and
$2\leq\add_{\mathcal H}(\F)\leq |{\mathcal H}|$, then
\[\max\{\add_{{\mathcal H}}(\F),\add_{{\mathcal H}}(\real^{\real}\setminus\F)\}
\leq \add^{*}_{{\mathcal H}}(\F).\]}
\proof We first show that $\add_{{\mathcal H}}(\real^{\real}\setminus\F)
\leq\add^{*}_{{\mathcal H}}(\F)$.   Let $F\subseteq{\mathcal H}$ be a witness to
the definition of $\add^{*}_{{\mathcal H}}(\F)$,  i.e., $|F|=\add^{*}_{{\mathcal
H}}(\F)$ and
\begin{equation*}
\left(\forall G\in [{\mathcal H}]^{<\add_{{\mathcal H}}(\F)}\right)
(\exists f\in
F)  (f+G\subseteq\F).
\end{equation*} Since $\add_{{\mathcal H}}(\F)\geq 2>1$, we see that $F$ also
satisfies
\begin{equation*} (\forall g\in{\mathcal H}) (\exists f\in F) (f+g\in\F).
\end{equation*} Since
$\F=\real^{\real}\setminus(\real^{\real}\setminus\F)$,  we see that
$\add_{\mathcal H}^*(\F)=|F|\geq\add_{{\mathcal
H}}(\real^{\real}\setminus\F)$.

We show $\add_{{\mathcal H}}(\F)\leq\add^{*}_{{\mathcal H}}(\F)$.   By way
of contradiction, assume that we have
$\add_{{\mathcal H}}(\F)>\add^{*}_{{\mathcal H}}(\F)$.   Then there is an
$F\subseteq{\mathcal H}$ such that
$|F|<\add_{{\mathcal H}}(\F)$  and
\begin{equation}\label{eq3:wart1}
\left(\forall G\in [{\mathcal H}]^{<\add_{{\mathcal H}}(\F)}\right)
(\exists f\in
F)  (f+G\subseteq\F).
\end{equation}   We claim that for each
$f\in F$ there is a $g_f\in{\mathcal H}$ such that $f+g_f\notin\F$.
Otherwise, there would be some $f\in F \subseteq{\mathcal H}$ such  that
$f+{\mathcal H}\subseteq\F$, which would imply that
$\add_{{\mathcal H}}(\F)=|{\mathcal H}|^+$, contradicting the  assumption that
$\add_{\mathcal H}(\F)\leq |{\mathcal H}|$.   Let $G=\{g_f\colon f\in F\}$.
Notice that $|G|\leq |F|<\add_{{\mathcal H}}(\F)$.   By (\ref{eq3:wart1}),
there is an $f\in F$ such that $f+G\subseteq\F$, in particular,
$f+g_f\in\F$; but this contradicts the choice of $g_f$.   Thus,
$\add_{{\mathcal H}}(\F)\leq\add^{*}_{{\mathcal H}}(\F)$.\qed

\section{The Results}
To state the theorems dealing with the super-additivities of the
families under consideration, we must first make a definition.  Let
$\F$  be a collection of subsets of $\real$.  We define the {\em
cofinality of $\F$} to be
\[\cof(\F)=\min\{|F|\colon F\subseteq\F\
\&\ (\forall M\in \F)(\exists N\in F)(M\subseteq N)\}.\] This cardinal
has been studied intensively for the case  when $\F$ is some ideal of
subsets of $\real$ \cite{BAJU}.   We now state the main theorems
and then discuss  their interpretations in terms of super-additivity
and additivity.

\noindent\thm{thm:1}{There exists a family $F\subseteq\swiat\cap\baire_1$  such
that
$|F|=\cof(\meager)$ and for any $H\in [\baire_{1}]^{<\omega}$  there is
an $f\in F$ such that $f+H\subseteq\dar\cap\quasi$.   Moreover, given
$\epsilon>0$, we may assume that
$\norm(f)\leq\sup\{\osc(g,x)\colon g\in H\ \&\ x\in\real\}+\epsilon$.}

\noindent\thm{thm:12}{There exists a family $F\subseteq\swiat\cap\baire_1$  such
that
$|F|=\cof(\meager)$ and for any $H\in [\cliq]^{<\omega}$ there is an
$f\in F$ such that $f+H\subseteq\pr\cap\quasi$.  Moreover, given
$\epsilon>0$ we may assume that
$\norm(f)\leq\sup\{\osc(g,x)\colon g\in H\ \&\ x\in\real\}+\epsilon$.}

The next theorem will be used to show that $\cof(\meager)$ is  actually
the smallest cardinal for which either Theorem~\ref{thm:1} or
Theorem~\ref{thm:12} remains true.

\noindent\thm{thm:3}{There is a family $F\subseteq\baire_1$ such that
$|F|=\cof(\meager)$ and
\begin{equation}\label{eq3:heav}
\left(\forall G\in [\cliq]^{<\cof(\meager)}\right)(\exists f\in F)
(f+G\subseteq (\real^{\real}\setminus(\quasi\cup\phc))).
\end{equation}} We also have a version of Theorem~\ref{thm:3} which will
allow us to say some things about Darboux quasi-continuous functions.

\noindent\thm{thm:31}{There is a family $F\subseteq\baire_1\cap\quasi$
such that
$|F|=\cof(\meager)$ and
\begin{equation}\label{eq3:heav1}
\left(\forall G\in [\cliq]^{<\cof(\meager)}\right)(\exists f\in F)
(f+G\subseteq (\real^{\real}\setminus\dar)).
\end{equation}}

The above theorems yield a large number of statements about the  additivities
and super-additivities of some of the families which  we are studying.

\noindent\cor{cor3:10}{ \
\begin{description}
\item[(i)]  For $\F\in\{\dar,\quasi,\dar\cap\quasi,\dar\cup\quasi\}$,  we
have \[\add_{\baire_1}(\real^{\real}\setminus\F)=
\add^*_{\baire_1}(\real^{\real}\setminus\F)=\add^*_{\baire_1}(\F)
=\cof(\meager).\]
\item[(ii)] For $\F\in\{\pr,\quasi,\pr\cap\quasi,\phc\cup\quasi\}$,  we
have \[\add_{\cliq}(\real^{\real}\setminus\F)=
\add^*_{\cliq}(\real^{\real}\setminus\F)=\add^*_{\cliq}(\F)=\cof(\meager).\]
\item[(iii)] $\add_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)=
\add^*_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)=
\add^*_{\quasi\cap\baire_1}(\dar)=\cof(\meager)$.
\end{description}}
\proof We show (i).  By containment, we have, using Corollary~\ref{cor3:1}
and  Proposition~\ref{prop:sup}(iii),
\begin{equation}\label{eq3:wwt}
\add^*_{\baire_1}(\dar\cap\quasi)\geq
\add^*_{\baire_1}(\F)\geq\add^*_{\baire_1}(\quasi\cup\dar)
\end{equation}  for $\F\in\{\dar,\quasi\}$.  Since
$\add_{\baire_1}(\dar\cap\quasi)=\omega$, Theorem~\ref{thm:1} implies
that
\begin{equation}\label{eq3:wt1}
\cof(\meager)\geq\add_{\baire_1}^*(\dar\cap\quasi)
\end{equation}
since, letting $F$ be as in Theorem~\ref{thm:1}, if
$G\in[\baire_1]^{<\omega}$,  then there is an $f\in F\subseteq\baire_1$
such that
$f+G\subseteq\dar\cap\quasi$.   By (\ref{eq3:wwt}) and (\ref{eq3:wt1})
together with Proposition~\ref{prop:51}, we  have
\begin{equation}\label{eq3:wwtt:1}
\cof(\meager)\geq\add^*_{\baire_1}(\quasi\cup\dar)\geq
\add_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi)).
\end{equation}
We claim that  Theorem~\ref{thm:3} implies that
%%%%%%%
\begin{equation}\label{eq3:wt2}
\add_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi))\geq\cof(\meager).
\end{equation}
To see the inequality, let $F$ be as in
Theorem~\ref{thm:3}.  Let
$G\in[\baire_1]^{<\cof(\meager)}$.  Since $\baire_1\subseteq\cliq$, there
is an $f\in F\subseteq\baire_1$ such that
$f+G\subseteq\real^{\real}\setminus (\dar\cup\quasi)$.  So the
inequality  holds. By (\ref{eq3:wwt}), (\ref{eq3:wt1}), (\ref{eq3:wwtt:1}),
and (\ref{eq3:wt2}) we have
\[\add^*_{\baire_1}(\F)=\add_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi))
=\cof(\meager)\]  for
$\F\in\{\dar,\quasi,\dar\cap\quasi,\dar\cup\quasi\}$.   Using
Proposition~\ref{prop:667}(iii), we have
\begin{equation}\notag
\add_{\baire_1}(\real^{\real}\setminus(\dar\cap\quasi))\geq
\add_{\baire_1}(\real^{\real}\setminus\F)\geq
\add_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi))=
\cof(\meager)
\end{equation}   for $\F\in\{\dar,\quasi\}$.  Theorem~\ref{thm:1} implies
that
$\add_{\baire_1}(\real^{\real}\setminus(\dar\cap\quasi))\leq\cof(\meager)$.
Thus,
\begin{equation}\label{eq3:wt3}
\add_{\baire_1}(\real^{\real}\setminus\F)=\cof(\meager)
\end{equation} for $\F\in\{\dar,\quasi,\dar\cap\quasi,\dar\cup\quasi\}$.
Using  (\ref{eq3:wt3}) and Proposition~\ref{prop:sup}(iii), we have, by
containments,
\begin{equation}\label{eq3:wt4}
\add^*_{\baire_1}(\real^{\real}\setminus(\dar\cap\quasi))\leq
\add^*_{\baire_1}(\real^{\real}\setminus\F)\leq
\add^*_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi))
\end{equation} for $\F\in\{\dar,\quasi\}$.   Since
$\add_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi))=\cof(\meager)$,
Theorem~\ref{thm:3} implies that
\begin{equation}\label{eq3:wt5}
\add^*_{\baire_1}(\real^{\real}\setminus(\dar\cup\quasi))\leq\cof(\meager).
\end{equation}
By Proposition~\ref{prop:51} and (\ref{eq3:wt3}),
$\add^*_{\baire_1}(\real^{\real}\setminus(\dar\cap\quasi))\geq
\add_{\baire_1}(\real^{\real}\setminus(\dar\cap\quasi))=\cof(\meager)$,
so we have
\begin{equation}\label{eq3:wt6}
\cof(\meager)\leq\add^*_{\baire_1}(\real^{\real}\setminus(\dar\cap\quasi)).
\end{equation} Thus, by (\ref{eq3:wt5}), (\ref{eq3:wt4}), and
(\ref{eq3:wt6}),  it follows that
$\add^*_{\baire_1}(\real^{\real}\setminus\F)=\cof(\meager)$  for every
$F\in\{\dar,\quasi,\dar\cap\quasi,\dar\cup\quasi\}$, which completes the
proof  of (i).

The proof of (ii) follows that of (i) except Theorem~\ref{thm:12} is
used instead of Theorem~\ref{thm:1}, and $\dar$ is replaced by
$\pr$.  One must also use the fact, see Proposition~\ref{prop:2}, that
$\dar\cap\quasi\subseteq\phc\cap\quasi=\pr\cap\quasi$.

We now prove (iii).  Since $\swiat\subseteq\quasi\subseteq\cliq$  and
$\add_{\quasi\cap\baire_1}(\dar)=\omega$,  Theorem~\ref{thm:1} implies
that
$\add_{\quasi\cap\baire_1}^*(\dar)\leq\cof(\meager)$.  By
Proposition~\ref{prop:51},
\begin{equation}\label{eq3:wt7}
\add_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)
\leq\add_{\quasi\cap\baire_1}^*(\dar)\leq\cof(\meager).
\end{equation}
Theorem~\ref{thm:31} implies that
$\add_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)\geq\cof(\meager)$;
so, by  (\ref{eq3:wt7}),
\[\add_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)
=\add_{\quasi\cap\baire_1}^*(\dar)=\cof(\meager).\]  Since
$\add_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)=\cof(\meager)$,
Theorem~\ref{thm:31} implies that
\[\add_{\quasi\cap\baire_1}^*(\real^{\real}\setminus\dar)\leq\cof(\meager),\]
and Proposition~\ref{prop:51} implies that
\[\add_{\quasi\cap\baire_1}^*(\real^{\real}\setminus\dar)\geq
\add_{\quasi\cap\baire_1}(\real^{\real}\setminus\dar)=\cof(\meager).\]
Thus,
$\add_{\quasi\cap\baire_1}^*(\real^{\real}\setminus\dar)=\cof(\meager)$
which completes the proof of (iii).\qed

The fact that $\add_{\baire_1}(\real^{\real}\setminus\dar)=\cof(\meager)$,
together with Proposition~\ref{prop:47}, implies  that the minimum number
of translations of $\baire_1\cap\dar$ by  Baire class one functions
required to cover Baire class one is $\cof(\meager)$.


We make heavy use of the following proposition throughout the remainder of
this paper.
\noindent\prop{prop:frem}{{\rm \cite[Theorem 3B]{FREM}}
$\cof(\meager)=\cof(\cnwd)$.}


\section{Proof of Theorems~\ref{thm:3} and \ref{thm:31}}

{\sc Proof of Theorem~\ref{thm:3}}.   By Proposition~\ref{prop:frem},
there exists
$\cnwd_0\subseteq\cnwd$ such that $|\cnwd_0|=
\cof(\meager)$ and $\cnwd_0$ satisfies
\begin{equation}\label{eq3:he1}
(\forall M\in\cnwd)(\exists K\in\cnwd_0)(M\subseteq K).
\end{equation}
For each $K\in\cnwd_0$, let $f_K=4\cdot\charf{K}$, notice $f_K\in\baire_1$.
Put $F=\{f_K\colon K\in\cnwd_0\}$.  Since
$|F|=\cof(\meager)$, it is enough for us to show that $F$ satisfies
(\ref{eq3:heav}).   Let $G\subseteq\cliq$ and $|G|<\cof(\meager)$.  We
find an
$f\in F$ such that $f+g\notin\quasi\cup\phc$ for every $g\in G$.
For each $g\in G$ let $A_g=\osc_1(g)$.  Note that by
Proposition~\ref{prop:aaa}, we  have
$A_g\in\cnwd$.   Since $|\{A_g\colon g\in
G\}|\leq |G|<\cof(\meager)$ and
$\cof(\meager)=\cof(\cnwd)$, there is a
$M\in\cnwd$ such that $M\setminus A_g\neq\emptyset$ for every
$g\in G$.  By (\ref{eq3:he1}), there is a $K\in\cnwd_0$ such that
$M\subseteq K$.   Notice that, since $M\subseteq K$, we have
$K\setminus A_g\neq\emptyset$ for every $g\in G$.   We claim that
$f_K+g\notin\quasi\cup\phc$ for each $g\in G$.   Fix $g\in G$.

Since $K\setminus A_g\neq\emptyset$, there is a $x_0\in K\setminus A_g$ and
an open interval $U$ such that $x_0\in U$ and $g[U]\subseteq
(g(x_0)-1,g(x_0)+1)$.  Since
$K\in\cnwd$, we may assume that $x_0$ is not a  bilateral  limit
point of $K$.  Without loss of generality, we may also assume that there is a
$\delta>0$ such that $J=(x_0,x_0+\delta)\subseteq U$ and
$J\cap K=\emptyset$.  Since $J\cap K=\emptyset$ implies
$f_K[J]=\{0\}$, we have $(f_K+g)[J]=g[J]\subseteq (g(x_0)-1,g(x_0)+1)$.
On the  other hand, $x_0\in K$ so $(f_K+g)(x_0)\geq
4+g(x_0)-1>g(x_0)+1$.   Thus, $f_K+g\notin\phc$.

We now show that $f_K+g\notin\quasi$.  Notice that
\[(f_K+g)(x_0)\in (g(x_0)+3,g(x_0)+5).\]   It is enough for us to show
that there is no non-empty open set
$W\subseteq U$ such that
$(f_K+g)[W]\subseteq (g(x_0)+3,g(x_0)+5)$.  For any non-empty open set
$W\subseteq U$, there is some
$x\in W\setminus K$.  Since $x\notin K$, we have
\[(f_K+g)(x)=0+g(x)\in (g(x_0)-1,g(x_0)+1).\]   So, $(f_K+g)[W]$ is not a
subset of $(g(x_0)+3,g(x_0)+5)$ for any non-empty open $W\subseteq U$.   Thus,
$f_K+g\notin\quasi$.\qed


To prove Theorem~\ref{thm:31}, it will be helpful to have some definitions
and  lemmas.  An open set $U\subseteq\real$ is said to be regular
provided that
$\interior(\cl(U))=U$.

\noindent\lem{lem:nice1}{The characteristic function of an open set $U$  is
quasi-continuous if and only if $U$ is regular.}
\proof
Assume $U$ is regular.  We show that $\charf{U}$ is
quasi-continuous.   If $x\in U$, then $x$ is clearly a point of continuity
and thus a point of quasi-continuity for $\charf{U}$.  So assume $x\notin
U$.  Let
$W$ be an open set such that $x\in W$.  By regularity, we have that
$x\notin\interior(\cl(U))$.   So, $W\setminus\cl(U)\neq\emptyset$ and
open.   It follows that, $\charf{U}(x)=0$ and
$\charf{U}[W\setminus\cl(U)]=\{0\}$; so, $\charf{U}$  is quasi-continuous.

Assume $\charf{U}$ is quasi-continuous.   We show that $U$ is regular.
Let $x\in\real\setminus U$.   Since $\charf{U}(x)=0$, quasi-continuity
implies that for  every open neighborhood $W$ of $x$ there is an open set
$\emptyset\neq V\subseteq W$ such that $\charf{U}[V]=\{0\}$. Since
$V\subseteq (W\setminus U)$ is non-empty and open, it follows that
$W\setminus\cl(U)\neq\emptyset$.  So,
$x\notin\interior(\cl(U))$.  Thus, $\interior(\cl(U))\subseteq U$.  The
containment
$U\subseteq\interior\cl(U)$ holds for any open set.  Thus, $U$  is
regular.\qed

\noindent\lem{lem:nice2}{Let $E\subseteq\real$ be a closed set which is nowhere
dense. Then, there is a regular open set $U$ such that
$E\subseteq\bd(U)$.}
\proof
Let $\{(a_n,b_n)\colon n\in\omega\}$ be the connected components of
$\real\setminus E$.   For each $n\in\omega$,
let $\langle x_{n,k}\rangle_{k\in\omega}$ and
$\langle y_{n,k}\rangle_{k\in\omega}$ be sequences of points in
$(a_n,b_n)$ such that $x_{n,0}=y_{n,0}$, $x_{n,k}\searrow a_n$,
and $y_{n,k}\nearrow b_n$.  Let
\begin{equation*}
U=\bigcup\{(x_{n,2k+1},x_{n,2k})\colon n,k\in\omega\}
\cup\bigcup\{(y_{n,2k},y_{n,2k+1})\colon n,k\in\omega\}.
\end{equation*}
It is now easily checked that $U$ has the desired
properties.\qed


\noindent{\sc Proof of Theorem~\ref{thm:31}.}   By
Proposition~\ref{prop:frem} there is a $\cnwd_0\subseteq\cnwd$ such that
$|\cnwd_0|=\cof(\meager)$ and
$\cnwd_0$ satisfies
\begin{equation}\label{eq3:he2} (\forall M\in\cnwd)(\exists
K\in\cnwd_0)(M\subseteq K).
\end{equation} By Lemma~\ref{lem:nice2}, we may find for each
$K\in\cnwd_0$ a regular open  set $U_K$ such that $K\subseteq \bd(U_K)$.
For each $K\in\cnwd_0$, let $f_K=4\cdot\charf{U_K}$.   By
Lemma~\ref{lem:nice1}, $f_K$ is quasi-continuous.  Moreover, $f_K$  is in
Baire class 1.  Put $F=\{f_K\colon K\in\cnwd_0\}$.  Since
$|F|\leq\cof(\meager)$, it  is enough for us to show that $F$ satisfies
(\ref{eq3:heav1}).   Let $G\subseteq\cliq$ and $|G|<\cof(\meager)$.  We
must find an
$f\in F$ such that $f+g\notin\dar$ for every $g\in G$.   For each $g\in
G$, let $A_g=\osc_1(g)$.  By Proposition~\ref{prop:aaa},
we have $A_g\in\cnwd$.  Since $|\{A_g\colon g\in
G\}|\leq |G|<\cof(\meager)=\cof(\cnwd)$,  there is a
$M\in\cnwd$ such that $M\setminus A_g\neq\emptyset$ for every
$g\in G$.  By (\ref{eq3:he2}), there is a $K\in\cnwd_0$ such that
$M\subseteq K$.   Notice that $K\setminus A_g\neq\emptyset$ for every
$g\in G$.   We claim that $f_K+g\notin\dar$ for each $g\in G$.  Fix $g\in
G$.   Let $x_0\in K\setminus A_g$.  Since $x_0\notin A_g$, there is an
open interval
$V$ such that $x_0\in V$ and $|g(x_0)-g(x)|<1$ for all $x\in V$.
Clearly, $g[V]\subseteq (g(x_0)-1,g(x_0)+1)$.  Note that
$\sup((f_K+g)[V])\geq 4+g(x_0)-1=g(x_0)+3$ since $x_0\in\cl(U_K)$.
Since $x_0\notin U_K$, we also have $\inf((f_K+g)[V])\leq g(x_0)+1$.
We claim that $g(x_0)+2\notin (f_K+g)[V]$ which will show that
$f_K+g\notin\dar$.   To see this, notice that if $x\in U_K\cap V$, then
$(f_K+g)(x)=g(x)+4\leq(g(x_0)-1)+4=g(x_0)+3$; and if
$x\in V\setminus U_K$, then
$(f_{K}+g)(x)=0+g(x)<g(x_0)+1$.  Thus, $f_K+g\notin\dar$. \qed


\section{Proof of Theorems~\ref{thm:1} and \ref{thm:12}} To prove
Theorems~\ref{thm:1} and \ref{thm:12} we must have an alternative
description of $\cof(\meager)$.  To this end, we introduce some new
notation.  For $S\subseteq\real$, we  define the bilateral closure of $S$
to be
\[\cl^*(S)=\{x\in\real\colon x\text{ is a bilateral limit point of S}\}.\]
For $F_1,F_2\in\cnwd$ and
$G\subseteq\real$, we write
$F_1\prec_{G} F_2$ provided that $F_1\subseteq\cl^*(F_2\cap G)$.   If
$G\subseteq\real$ and
$\Theta,\Psi\in \cnwd^{\omega}$, we write
$\Theta\prec_{G}\Psi$ provided that $\Theta(n)\prec_{G}\Psi(n)$  for
every $n\in\omega$.  In what follows we will  call $K\in\cnwd$ a {\em
Cantor set} provided that $K$ is dense in itself.


\noindent\lem{lem:12}{Let $E\subseteq\real$ be a closed set nowhere dense and
$G\subseteq\real$ be a dense $G_{\delta}$-set.   Then there is a Cantor
set $N$ such that $N\setminus E\subseteq G$ and
$E\prec_G N$.}
\proof  Let $\{(a_n,b_n)\colon n\in\omega\}$ be the connected components of
$\real\setminus E$.   For each $n\in\omega$,
let $\langle P_{n,k}\rangle_{k\in\omega}$ and
$\langle Q_{n,k}\rangle_{k\in\omega}$ be sequences of perfect sets in
$(a_n,b_n)\cap G$ so that
$\lim_{k\to\infty}\sup\{|x-a_n|\colon  x\in P_{n,k}\}=0$ and
$\lim_{k\to\infty}\sup\{|x-b_n|\colon  x\in Q_{n,k}\}=0$.  Let
\[N=\cl\left[\left(\bigcup\{P_{n,k}\colon n,k\in\omega\}\right)
\cup\left(\bigcup\{Q_{n,k}\colon n,k\in\omega\}\right)\right].\]   It is easily
checked that $N$ has the desired properties.\qed

We are now ready to give our description of $\cof(\meager)$.
\noindent\lem{lem:4}{$\cof(\meager)=\kappa$ where
\[\kappa=\min\{|F|\colon F\subseteq \cnwd^{\omega}\ \&\  (\forall
G\in\dense)(\forall\Theta\in \cnwd^{\omega}) (\exists\Psi\in
F)(\Theta\prec_{G}\Psi)\}.\]}
\proof
We first show that $\cof(\meager)\leq\kappa$.   Let $F\subseteq
\cnwd^{\omega}$ be such that $|F|=\kappa$ and
\begin{equation*} (\forall G\in\dense)(\forall\Theta\in (\cnwd)^{\omega})
(\exists\Psi\in F)(\Theta\prec_{G}\Psi).
\end{equation*}
Let $\cnwd_1=\{\Psi(0)\colon\Psi\in F\}$.  Note that
$|F|\leq\kappa$.   Let $K\in\cnwd$.  Define $\Theta\in \cnwd^{\omega}$ so
that
$\Theta[\omega]=\{K\}$.  There is a $\Psi\in F$ such that
$\Theta\prec_{\real}\Psi$.  In particular,
$K=\Theta(0)\subseteq\Psi(0)$  since $\Psi(0)$ is closed and
$\Theta(0)\prec_{\real}\Psi(0)$.  So, for every $K\in\cnwd$
there is a
$K_1\in\cnwd_1$ such that $K\subseteq K_1$.   Thus,
$\cof(\cnwd)\leq\kappa$.  Hence, by Proposition~\ref{prop:frem},
$\cof(\meager)\leq\kappa$.

We now work for the other inequality.   Let $\cnwd^*$ and $\dense^*$
stand for the collections of subsets in
$(0,1)$ which are closed nowhere dense in $(0,1)$ and co-meager  in
$(0,1)$, respectively.  Let
\[\kappa_1=\min\{|F|\colon F\subseteq (\cnwd^*)^{\omega}\ \&\  (\forall
G\in\dense^*)(\forall\Theta\in (\cnwd^*)^{\omega}) (\exists\Psi\in
F)(\Theta\prec_{G}\Psi)\}.\]    Notice that, by homeomorphism,
$\kappa_1=\kappa$.   Recalling the definition of $\cof(\meager)$ and
considering complements,  there is a collection $D\subseteq\dense$  such
that $|D|=\cof(\meager)$ and
\begin{equation}\label{e4} (\forall G\in\dense)(\exists H\in
D)(H\subseteq G).
\end{equation} By Proposition~\ref{prop:frem}, there is a family
$F\subseteq\cnwd$ such that $|F|=\cof(\meager)$ and
\begin{equation}\label{e5a} (\forall M\in \cnwd)(\exists N\in
F)(M\subseteq N).
\end{equation} For each $M\in F$ and $H\in D$, define
$\Psi_M^H\subseteq (\cnwd^*)^{\omega}$ as follows.  By Lemma \ref{lem:12}
there is a $K_M^H\in\cnwd$  such that
$M\prec_{H} K_M^H$.  We define for each $n\in\omega$
\begin{equation}\label{eq3:wart3}
\Psi_M^H(n)=(K_M^H\cap(n,n+1))-n.
\end{equation}    Let
$F^*=\{\Psi_M^H\colon M\in F\text{ and }H\in D\}$.  Since
$|D|=\cof(\meager)=|F|$, it follows that
$|F^*|\leq\cof(\meager)$.   We will be done if we show that $F^*$
satisfies
\begin{equation}\label{e5} (\forall G\in\dense^*)(\forall\Theta\in
(\cnwd^*)^{\omega})(\exists\Psi\in F^*)(\Theta\prec^{*}_{G}\Psi).
\end{equation} So let $G\in\dense^*$ and $\Theta\in (\cnwd^*)^{\omega}$.
We find a
$\Psi\in F^*$ such that $\Theta\prec_{G}\Psi$.  Let
$G_1=\left(\bigcup_{n\in\omega}(n+G)\right)\cup (-\infty,0)$ and notice that
$G_1\in\dense$.  Define
$N=\cl\left(\bigcup_{n\in\omega}(n+\Theta(n))\right)$ and notice that
$N\in\cnwd$.  Pick $H\in D$ and $M\in F$ so that
$H\subseteq G_1$ and $N\subseteq M$.  We claim that
$\Theta\prec_{G}^*\Psi_M^H$.   Let $K_M^H$ be as in the definition of
$\Psi_M^H$; in particular, we have
$M\prec_{H} K_M^H$.  Since $H\subseteq G_1$, we have $M\prec_{G_1}
K_M^H$.   Let $k\in\omega$  be arbitrary and
$p\in\Theta(k)$.  Then
$p+k\in\Theta(k)+k\subseteq N\cap (k,k+1)$.  Since $N\subseteq M$ and
$M\prec_{G_1} K_M^H$, we have
$\{p+k\}\prec_{G_1} K_M^H\cap (k,k+1)$.  But then,
$\{p\}\prec_{G} (K_M^H\cap (k,k+1))-k=\Psi_M^H(k)$.   So,
$\Theta(k)\prec_G\Psi^{H}_{M}(k)$.  Since
$k$ was arbitrary, we have $\Theta\prec_{G}^*\Psi_M^K$.  Thus,
(\ref{e5}) is satisfied.\qed

We now work to construct a family $F\subseteq\swiat\cap\baire_1$  of
cardinality $\cof(\meager)$ that will work for both  Theorems~\ref{thm:1}
and \ref{thm:12}.
\noindent\lem{lem:2b}{Let $\rho\in (0,+\infty]$, $K,M\in\cnwd$, and
$F$ be a Cantor set such that $M\subseteq\cl^*(F\setminus M)$ and
$F\cap K\subseteq M$.  Given an $f\in\swiat\cap\baire_1$ with
\begin{description}
\item[($p_1$)] $\real\setminus M\subseteq\cont(f)$ and
\item[($p_2$)] if $x\in M\setminus\cont(f)$ then $\acc_{K\setminus M}
(f,x)\supseteq(f(x)-\rho,f(x)+\rho)$, then
\end{description}
one may find a $g\in\baire_1$ such that
$g+f\in\swiat\cap\baire_1$ and
\begin{description}
\item[($a$)] $\real\setminus M=\cont(g+f)$
\item[($b$)] $g[M\cup K]=\{0\}$
\item[($c$)] $\acc_{(F\cup K)\setminus M}
((g+f),x)\supseteq((f+g)(x)-\rho,(g+f)(x)+\rho)$  for each $x\in M$
\item[($d$)] $\norm(g)\leq 2\rho$.
\end{description} Moreover, if $f=\charf{\emptyset}$, we may improve ($d$)
to
$\norm(g)\leq\rho$.}
\proof  Let $I$ be a connected component of the complement of $M$.   We
show how to define
$g$ on $I$.   First assume that $I$ has finite endpoints $a<b$.   We show
how to define $g$ on $(a,(b+a)/2]$.  If $a\notin\cont(f)$, then  let
$g[(a,(b+a)/2]]=\{0\}$.  Suppose that $a\in\cont(f)$.   Since $F$ is a
Cantor set, $f|_{I\cup\{a\}}$ is continuous, and $a\in M$ we may find a
sequence of pairwise disjoint Cantor sets $\{F_n\}_{n\in\omega}$ such
that
\begin{description}
\item[(1)] $\bigcup_{n\in\omega}F_n\subseteq (a,(b+a)/2)\cap F$,
\item[(2)] $f|_{F_n}$ is continuous for each $n\in\omega$,
\item[(3)] $\lim_{n\to\infty}\sup\{|x-a|\colon x\in F_n\}=0$, and
\end{description}
We must
consider two cases.  First, assume that $\rho$ is finite.   For every
$n\in\omega$ large enough, we have
$\sup\{|f(a)-f(x)|\colon x\in F_n\}<\rho$.  For all such $n$ there is a
continuous function
$g_n\colon F_n\to\real$ such that
$(g_n+f)[F_n]=[f(a)-\rho,f(a)+\rho]$ and
$\norm(g_n)\leq 2\rho$.  For small values of $n$, we define $g_n$ so  that
$g_n[F_n]=\{0\}$. Notice that if $f=\charf{\emptyset}$, we may for every
$n\in\omega$ pick a $g_n$ such that
$(g_n+f)[F_n]=[-\rho,\rho]$ and $\norm(g_n)\leq\rho$.   By the Tietze
extension theorem \cite[p.127]{KUR}, there is a continuous function
$g\colon (a,(b+a)/2]\to\real$  such that
$g|_{F_n}=g_n$ for every $n\in\omega$,
$g[K\cup\{(b+a)/2\}]=\{0\}$, and $\norm(g)\leq 2\rho$  (or
$\norm(g)\leq\rho$).   In the case when $\rho=+\infty$, we define $g$ on
$(a,(b+a)/2]$ as we did when $\rho$ was finite, except we define
$g_n\colon F_n\to\real$ so that $(g_n+f)[F_n]=[f(a)-n-1,f(a)+n+1]$  for
every $n\in\omega$.   We may do the symmetrical construction to define
$g$ on $[(b+a)/2,b)$.    This completes the construction of $g$  for when
$I$ has two finite endpoints.

If $I$ has one finite endpoint, $p$ repeat
the construction above for $p$.   For example if $I=(p,+\infty)$, the
desired continuous function would be of the form $g$, above, for points
close to $p$ and constantly equal to zero otherwise.   We do one of the
constructions mentioned above for each interval of the complement of $M$
and let
$g|_M=\{0\}$ to complete the construction of
$g$.

We now show that $g$ has the desired properties.   First, notice that
($b$) and ($d$) (or $\norm(g)\leq\rho$)  are immediate from the
construction.  To show $g\in\baire_1$, it is enough to show that
$g^{-1}((r,+\infty))$ and $g^{-1}((-\infty,r))$ are $F_{\sigma}$-sets
for every $r\in\real$ \cite[p.373]{KUR}. Since $\real\setminus
M\subseteq\cont(g)$, it follows that
$g^{-1}((r,+\infty))\cap(\real\setminus M)$ is open and thus an
$F_{\sigma}$-set.  By ($b$), if $r<0$, then
$g^{-1}((r,+\infty))=(g^{-1}((r,+\infty))\cap(\real\setminus M))\cup M$,
which is an $F_{\sigma}$-set.  By ($b$), if $r\geq 0$, then
$g^{-1}(r,+\infty))=g^{-1}((r,+\infty))\cap(\real\setminus M)$,
which is an
$F_{\sigma}$-set.  The argument that $g^{-1}((-\infty,r))$ is an
$F_{\sigma}$-set is similar.   We show that ($c$) holds.  Let $x\in M$.
If $x\in M\setminus\cont(f)$, then  we are done by ($b$) and ($p_2$).  So,
we may now assume that $x\in\cont(f)\cap M$.   To complete the proof of
($c$), we only show that
\begin{equation}\label{eq3:tew}
\acc^{+}_{F\setminus M}(g+f,x)\supseteq((g+f)(x)-\rho,(g+f)(x)+\rho)
\end{equation}
since the containment
$\acc^{-}_{F\setminus M}\supseteq((g+f)(x)-\rho,(g+f)(x)+\rho)$ follows by a
similar argument and $F\setminus M\subseteq (K\cup F)\setminus M$.   It
is clear from the construction and (b) that if $x\in M$ is the left hand
endpoint of a complementary interval of $M$, then (\ref{eq3:tew}) holds.
So, assume that $x$
is not the left-hand endpoint of any complementary interval.   Then we
may find a sequence $\{x_n\}_{n\in\omega}$ of left-hand endpoints of
complementary intervals with the property that
$\lim_{n\in\omega}x_n=x$.  Since $x\in\cont(f)$, we also have that
$\lim_{n\in\omega}f(x_n)=f(x)$.  It now follows by the fact that
(\ref{eq3:tew})
holds for left hand endpoints and
the fact that $g[M]=\{0\}$ that \[\acc^{+}_{F\setminus
M}(g+f,x)\supseteq((g+f)(x)-\rho,(g+f)(x)+\rho).\]   Thus, (\ref{eq3:tew})
holds, and ($c$) is established.    We show that ($a$) holds.  It is clear
from the construction  that $\real\setminus M\subseteq\cont(g)$.  So by,
($p_1$) we have
$\real\setminus M\subseteq\cont(f+g)$.  By ($c$), we have
$\cont(f+g)\subseteq\real\setminus M$.  Thus, ($a$) holds.

It remains to be shown that $g+f\in\baire_1\cap\swiat$.   We first show
that $g$ is peripherally continuous.  Let $x\in\real$.   If $x\notin M$,
then by ($a$) and ($p_1$), $x$ is a point of continuity for
$g+f$.  Thus, $x$ is a point of  peripheral continuity for $g+f$.  If
$x\in M$, then $x$ is a point of  peripheral continuity for $g+f$ by
($c$).  Thus, $g+f$ is  peripherally continuous and so, by
Proposition~\ref{prop:2}(i), Darboux.  We now work to show that
$g+f\in\baire_1\cap\swiat$.  Let $[a,b]$ be an arbitrary  non-trivial
interval.  If $c$ is some number strictly between
$(g+f)(a)$ and $(g+f)(b)$, then since $g+f$ is Darboux, there is some
$x\in (a,b)$ such that $(g+f)(x)=c$.  If $x$ is a point of continuity,
then we are done.  If $x$ is not a point of continuity, then $x\in M$; so,
$\acc_{(F\cup K)\setminus
M}(g+f,x)\supseteq((g+f)(x)-\rho,(g+f)(x)+\rho)$.   It follows that there
is some
$p\in ((F\cup K)\setminus M)\cap(a,b)$  such that $(g+f)(p)=c$, but
$((F\cup K)\setminus M)\subseteq\cont(f+g)$ by  ($a$).   Therefore,
$g+f\in\swiat\cap\baire_1$. \qed


\noindent\lem{lem:3}{Let $\langle E_n \rangle_{n\in\omega}$ be a  sequence of
nowhere dense closed subsets of $\real$ and for each
$n\in\omega$ put $E^*_n=E_n\setminus\bigcup_{k<n}E_k$.   Given $\tau\in
(0,+\infty]$ and $0<\epsilon\leq\tau$, there is a $f\in\swiat\cap\baire_1$
such that
\begin{description}
\item[(i)] $\cont(f)=\real\setminus\bigcup_{n\in\omega}E_n$;
\item[(ii)] $f|_{E^*_n}$ is continuous for each $n\in\omega$ and
$f[E_0]=\{0\}$;
\item[(iii)] if $x\in E_0$, then
$\acc_{\cont(f)}(f,x)\supseteq (-\tau,+\tau)$;
\item[(iv)] if $x\in E_{n}$ and $n>0$, then
$\acc_{\cont(f)}(f,x)\supseteq  (f(x)-\epsilon/2^n,f(x)+\epsilon/2^n)$; and
\item[(v)] $\norm(f)\leq\tau+2\epsilon$.
\end{description}}
\proof Let $E=\bigcup_{n\in\omega}E_n$ and
$G=\real\setminus E$.  Clearly, $G$ is a dense
$G_{\delta}$-set. We now construct a sequence $\langle
f_n\in\baire_{1}\cap\swiat\rangle_{n\in\omega}$  such that
$f=\lim f_n$ will have the desired properties.  Since $E_0$  is nowhere
dense, we may, by Lemma~\ref{lem:12}, find a Cantor set $F^{*}_0$  such that
$E_0\subseteq\cl^*(F^{*}_0\cap G)$ and $F_0^*\setminus E_0\subseteq G$.  By
Lemma~\ref{lem:2b} used with $M=E_0$,
$K=\emptyset$, $F=F^{*}_0$, $\rho=\tau$ and
$f=\charf{\emptyset}$, we may construct a
$g_0\in\swiat\cap\baire_1$ such that
\begin{description}
\item[($a_0$)] $\real\setminus E_0=\cont(g_0)$,
\item[($b_0$)] $g_0[E_0]=\{0\}$,
\item[($c_0$)] $\acc_{F^{*}_0\cap G}(g_0,x)\supseteq(-\tau,+\tau)$  for each
$x\in E_0$, and
\item[($d_0$)] $\norm(g_0)\leq\tau$.
\end{description}
For the inductive step, assume we have constructed  a
sequence $\{f_k\}_{k=0}^{n}$ and a sequence
$\{F_k\}_{k=0}^n$ of Cantor sets such that $f_0=g_0$ and $F_0=F^{*}_0$; and
for $0<k\leq
n$ we have $f_k\in\swiat\cap\baire_1$ and
\begin{description}
\item[($a_k$)] $\cont(f_k)=\real\setminus\bigcup_{i\leq k} E_i$,
\item[($b_k$)] $f_k|_{E_i\cup F_i}=f_i|_{E_i\cup F_i}$  for all $i<k$,
\item[($c_k$)] $\acc_{F_k\cap G}(f_k,x)\supseteq
(f_k(x)-\epsilon/2^k,f_k(x)+\epsilon/2^k)$ for every
$x\in\bigcup_{l\leq k} E_l$,
\item[($d_k$)] $\norm(f_k-f_{k-1})\leq\epsilon/2^{k-1}$,
\item[($e_k$)]  $F^{*}_{n+1}\setminus\bigcup_{k\leq n+1}E_k\subseteq G$.
\end{description} We find a Cantor set $F_{n+1}$ and
$f_{n+1}\in\swiat\cap\baire_1$ so that ($a_{n+1}$), ($b_{n+1}$),
($c_{n+1}$), ($d_{n+1}$), and ($e_{n+1}$) are all satisfied.  Since
$G_1=G\setminus\left((\bigcup_{k\leq n}F_k)\cup (\bigcup_{k\leq
n+1}E_{k})\right)$  is a dense $G_{\delta}$-set, there is, by
Lemma~\ref{lem:12}, a Cantor set $F^{*}_{n+1}$  such that
\begin{equation}\label{eq3:kle}
\bigcup_{k\leq n+1}E_k\subseteq\cl^*(F^{*}_{n+1}\cap G_1)
\text{ and }F^{*}_{n+1}\setminus\bigcup_{k\leq n+1}E_k\subseteq G_1.
\end{equation}
We claim that $f_n$ with $M=(\bigcup_{k\leq n+1}E_{k})$,
$K=(\bigcup_{k\leq n}F_k)$, and $F=F^{*}_{n+1}$ satisfy the conditions of
Lemma~\ref{lem:2b} with
$\rho=\epsilon/2^{n+1}$.  To  see ($p_1$) and ($p_2$) are satisfied, notice
that $f_n$ satisfies ($a_n$)  and ($c_n$), respectively.  Also notice
that, by (\ref{eq3:kle}), we have
$M\subseteq \cl^*(F\setminus M)$ and $F\cap K\subseteq M$.   Applying
Lemma~\ref{lem:2b} and letting
$S=(F_{n+1}^{*}\cup\bigcup_{k\leq n}F_k)\setminus\bigcup_{k\leq n+1}E_k$,
we may find a $g\in\baire_1$ such that $g+f_n\in\swiat\cap\baire_1$ and
\begin{description}
\item[($a_{n+1}^{*}$)] $\real\setminus\bigcup_{k\leq n+1}
E_{k}=\cont(f_n+g)$,
\item[($b_{n+1}^{*}$)] $g[ (\bigcup_{k\leq n}F_k)\cup(\bigcup_{k\leq
n+1}E_{k})]=\{0\}$,
\item[($c_{n+1}^{*}$)] $\acc_{S}(g+f_n,x)
\supseteq((g+f_n)(x)-\epsilon/2^{n+1},(g+f_n)(x)+\epsilon/2^{n+1})$ for
each
$x\in\bigcup_{k\leq n+1}E_{k}$, and
\item[($d_{n+1}^{*}$)] $\norm(g)\leq\epsilon/2^{n}$.
\end{description}

We claim that $f_{n+1}=f_n+g$ and
$F_{n+1}=F^{*}_{n+1}\cup(\bigcup_{k\leq n} F_k)$  are as desired.  Since
$f_{n+1}-f_n=g$ and $\norm(g)\leq\epsilon/2^{n}$, we have  ($d_{n+1}$).
We show that ($b_{n+1}$) is satisfied.   If $k<n+1$, then, by
($b_{n+1}^{*}$), $f_n|_{F_k\cup E_k}=f_{n+1}|_{F_k\cup E_k}$.  So by
($b_{n}$), we have $f_k|_{F_k\cup E_k}=f_{n+1}|_{F_k\cup E_k}$.  So
($b_{n+1}$) holds.  We now work to establish ($c_{n+1}$).  Let
$x\in\bigcup_{k\leq n+1} E_{k}$.  We consider two cases.  First, assume
$x\notin\cont(f_n)$.  By ($a_n$), there is a $k\leq n$ such that
$x\in E_k$.  By ($b_{n+1}$), we have
$f_k|_{F_k\cup E_k}=f_{n+1}|_{F_k\cup E_k}$.  This together with  ($c_k$)
and $F_k\subseteq F_{n+1}$ implies that
$\acc_{F_{n+1}\cap G} (f_{n+1},x)\supseteq (f_{n+1}(x)-\epsilon/2^k,
f_{n+1}(x)+\epsilon/2^k)$ if $k>0$; and, when $k=0$, we have
$\acc_{F_{n+1}\cap G} (f_{n+1},x)\supseteq
(f_{n+1}(x)-\tau,f_{n+1}(x)+\tau)$.  In  both cases we have
$\acc_{F_{n+1}\cap G} (f_{n+1},x)\supseteq (f_{n+1}(x)-\epsilon/2^{n+1},
f_{n+1}(x)+\epsilon/2^{n+1})$ since $\epsilon\leq\tau$.  We now
consider the case when
$x\in\cont(f_n)$.  By ($a_n$), we know that $x\in E_{n+1}$ so, by
($c_{n+1}^{*}$), $\acc_{S}(f_{n+1},x)
\supseteq(f_{n+1}(x)-\epsilon/2^{n+1},f_{n+1}(x)+\epsilon/2^{n+1})$.
Since $S\subseteq G_1\cap F_{n+1}\subseteq G\cap F_{n+1}$, we have
($c_{n+1}$).  It follows from (\ref{eq3:kle}) and ($e_{n}$) that
($e_{n+1}$) holds.
Finally, notice that ($a_{n+1}$) and follows immediately from
($a_{n+1}^*$).   This completes the inductive step.  We now let
$f=\lim_{n\to\infty} f_n$.

We show that $f$ satisfies (i)-(v).  We first  note that $f$ is the
uniform limit of
$\langle f_n \rangle_{n\in\omega}$ by ($d_n$).  Notice that ($d_n$)
together with ($a_0$) also yields (v).   We show (i).  If $x\notin E$
then, for every
$n\in\omega$ we have, by ($a_{n}$), $x\in\cont(f_n)$.   So by uniformity
of the convergence, $x\in\cont(f)$.  Thus,
$(\real\setminus E)\subseteq\cont(f)$.   Let $x\in E$.  There is an
$n\in\omega$ such that $x\in E_n$.   By $(c_n)$, $f_n|_{F_n}$ is
discontinuous at $x$.  Since $(b_m)$  holds for all $m>n$, we have that
$f|_{F_n}$ is discontinuous at $x$.   Thus, $x\notin\cont(f)$.  So
$\cont(f)\subseteq (\real\setminus E)$,  which establishes (i).   To see
that (ii) holds let $n\in\omega$.  By ($b_0$)
$f_0|_{E_0}$ is constant.  By ($b_m$) for $m>0$, we  have
$f|_{E_0}=f_0|_{E_0}$; so, $f|_{E_0}$ is continuous.  If $n\geq 0$  then, by
($a_n$), $E_{n+1}^*\subseteq\cont(f_n)$; so,
$f_n|_{E^*_{n+1}}$ is continuous.  By ($b_m$) for $m>n$, we  have
$f|_{E_{n+1}^*}=f_n|_{E_{n+1}^*}$; so, $f|_{E^*_{n+1}}$ is continuous.
Thus, (ii) holds. We show (iii) and (iv) hold.  Assume
$n>0$ and let $x\in E_n$.   We have, by ($c_n$) of the
construction, that $\acc_{F_{n}\cap G}(f_n,x)\supseteq
(f_{n}(x)-\epsilon/2^n,f_{n}(x)+\epsilon/2^n)$.  It follows, by ($b_m$)
for $m>n$, that
$\acc_{F_{n}\cap G}(f,x)\supseteq(f(x)-\epsilon/2^n,f(x)+\epsilon/2^n)$.
Using (i), we have $G=\cont(f)$ so
$\acc_{\cont(f)}(f,x)\supseteq(f(x)-\epsilon/2^n,f(x)+\epsilon/2^n)$.
Thus, (iv) holds.  A similar argument shows that (iii) holds.   We must
now show that $f\in\swiat\cap\baire_1$.  Note that since the uniform limit of
Darboux Baire 1 functions is again Darboux Baire 1
\cite[p.72]{BCW},
$f\in\dar\cap\baire_1$.  Let $a<b$ and assume without loss of generality
that $f(a)<f(b)$.   Let $f(a)<w<f(b)$ be arbitrary.  Since $f\in\dar$,
there is a $z\in (a,b)$ such that $f(z)=w$.  If $z\in\cont(f)$  then we
are done.  So, assume that
$z\notin\cont(f)$.  Since $\real\setminus\cont(f)=E$, there is an 0
$n\in\omega$ such that $z\in E_n$.  By (iii) or (iv), we may find an
$x\in\cont(f)\cap (a,b)$ such that $f(x)=w$.  Thus,
$f\in\swiat\cap\baire_1$.
\qed

To simplify the proof of Theorems~\ref{thm:1} and \ref{thm:12}, we
introduce another lemma.
\noindent\lem{lem:suff}{Let $f\in\real^{\real}$.
\begin{description}
\item[(a)] If $f\in\baire_1$ and $f(x)\in\wacc_{\cont(f)}(f,x)$  for
every $x\in\real$, then $f\in\quasi\cap\dar$.
\item[(b)] If $f\in\cliq$ and $f(x)\in\wacc_{\cont(f)}(f,x)$  for every
$x\in\real$, then $f\in\quasi\cap\pr$.
\item[(c)] If $g\in\real^{\real}$, $x\in\cont(g)$, and
$\cont(f)\subseteq\cont(g)$, then
\[g(x)+\wacc_{\cont(f)}(f,x)\subseteq\wacc_{\cont(f+g)}(f+g,x).\]
\item[(d)] If $x\in\cont(f)$, then $x$ is a point of quasi-continuity  and
peripheral continuity of $f$.
\end{description}}
\proof We show (a) and (b).  By (i) and (iv) of Proposition~\ref{prop:2},
it enough for us to show that if $f(x)\in\wacc_{\cont(f)}(f,x)$ for
every $x\in\real$, then $x\in\quasi\cap\phc$.  Let $x\in\real$.   Since
$f(x)\in\wacc_{\cont(f)}(f,x)$, there is a bilateral sequence
$\{x_k\}_{k\in\omega}$ such that $x_k\in\cont(f)$,
$\lim_{k\to\infty}x_k=x$, and $\lim_{k\to\infty}f(x_k)=f(x)$.   Clearly,
$x$ is a point of peripheral continuity.  By \cite{GRNAT},
$x$ is a point of quasi-continuity.

We show (c).  Let $r\in g(x)+\wacc_{\cont(f)}(f,x)$.  Then, there is  a
bilateral sequence $\{x_k\}_{k\in\omega}$ such that $x_k\in\cont(f)$,
$\lim_{k\to\infty}x_k=x$, and $\lim_{k\to\infty}f(x_k)=r-g(x)$.  Since
$x\in\cont(g)$, we have $\lim_{k\to\infty}g(x_k)=g(x)$.  Now
$\lim_{k\to\infty}(f+g)(x_k)=r$ and
$x_k\in\cont(f)\subseteq\cont(g)$ for every $k\in\omega$.  Thus,
$r\in\wacc_{\cont(f+g)}(f+g,x)$.

We leave (d) without proof as it follows immediately from the
definitions.  \qed



\bigskip

\noindent {\sc Proof of Theorem \ref{thm:1}.}   By Lemma \ref{lem:4}
there exists $F^*\subseteq \cnwd^{\omega}$  such that
$|F^*|=\cof(\meager)$ and
\begin{equation}\label{e7} (\forall G\in\dense)(\forall\Theta\in
\cnwd^{\omega}) (\exists\Psi\in F^*)(\Theta\prec_{G}\Psi).
\end{equation}
For each $\Psi\in F^*$, $q\in\rational^+\cup\{\infty\}$,
and
$n\in\omega$, define
$f_{\Psi}^{q,n}\in\swiat\cap\baire_1$ to be the function  constructed in
Lemma \ref{lem:3} with
$E_k=\Psi(k)$, $\tau=q$, and $\epsilon=\min\{\tau,2^{-n}\}$.   Let
$F=\{f^{q,n}_{\Psi}\colon q\in\rational^+\cup\{\infty\}\ \&\ n\in\omega\
\&\ \Psi\in F^*\}$; since
$\cof(\meager)>\omega$, we see that
$|F|=\cof(\meager)=|F^*|$.

We show that $F$ is as desired.  That is to say, we show that for  every
$\epsilon>0$ and $H\in[\baire_1]^{<\omega}$ there is an $f\in F$  with
the property that
$f+H\subseteq\dar\cap\quasi$ and
$\norm(f)\leq\max(\{\osc(g,x)\colon x\in\real\ \&\ g\in H\})+\epsilon$.
Let $H\in[\baire_1]^{<\omega}$ and $\epsilon>0$.  If
$\max\{\osc(g,x)\colon x\in\real\ \&\ g\in H\}=+\infty$, then  let
$\tau=+\infty$.  In the other case, we pick
$\tau\in\rational\cap (0,+\infty)$ such that
\begin{equation}\label{eq3:afinal}
0<\tau-\max\{\osc(g,x)\colon x\in\real\ \&\ g\in H\}<2^{-m}
\end{equation}
where
$m\in\omega$ is such that $2^{-m}<\min\{\tau,\epsilon/3\}$.   Define
$\Theta_H\in
\cnwd^{\omega}$ so that
\[\Theta_H(n)=\bigcup_{g\in H}\osc_{n+m+2}(g)\] for each
$n\in\omega$.  There is a $\Psi\in F^*$ such that $\Theta_H\prec_G\Psi$
where
$G\!=\!\bigcap_{g\in H}\cont(g)$.  We claim that
$f=f_{\Psi}^{\tau,m}\in F$ is the desired function.  It follows from (v)
of Lemma~\ref{lem:3} and  the choice of $m$ that
$\norm(f)\leq\tau+2\epsilon/3$.  By our choice of $\tau$ and $m$, we have
\[\tau+2\epsilon/3 \leq\max\{\osc(g,x)\colon x\in\real\ \&\ g\in
H\}+\epsilon/3+2\epsilon/3.\]  Thus, $\norm(f)\leq\max\{\osc(g,x)\colon
x\in\real\text{ and } g\in H\}+\epsilon$.   Fix $g\in H$.   We show that
$f+g\in\dar\cap\quasi$.  By Lemma~\ref{lem:suff}(a), we only have to show
that for every $x\in\real$
\begin{equation}\label{eq3:nuff}
(f+g)(x)\in\wacc_{\cont(f+g)}(f+g,x).
\end{equation}

Before proving (\ref{eq3:nuff}) we note that
\begin{equation}\label{eq3:300}
\cont(f)\subseteq\cont(g)
\end{equation}
since by (i) of Lemma~\ref{lem:3},
$\cont(f)=\real\setminus\bigcup_{i\in\omega}
\Psi(i)\subseteq
\real\setminus\bigcup_{i\in\omega}\Theta_{H}(i)\subseteq\cont(g)$.

Let $x\in\real$.  If $x\in\cont(g)$, then, by (\ref{eq3:300}), (c) of
Lemma~\ref{lem:suff}, and the fact that
$f(x)\in\wacc_{\cont(f)}(f,x)$ (Lemma~\ref{lem:3}(iii) and (iv))
we have
\[(f+g)(x)\in\wacc_{\cont(f+g)}(f+g,x).\]
If $x\notin\cont(g)$, then, by
(\ref{eq3:300}),
$x\notin\cont(f)$.  By (i) of Lemma~\ref{lem:3}, there is a minimal number
$n\in\omega$ such that
$x\in\Psi(n)$.  It is also true, since $\Theta_{H}(k)\subseteq\Psi(k)$
for every $k\in\omega$, that $n$ is less than or equal to the minimal
number $p$ such that
$x\in\Theta_H(p)$.   Since $p$ is minimal and $n\leq p$, we have in
the case when $n>0$ that
$x\notin\Theta_H(n-1)\supseteq\osc_{n+m+1}(g)$.   Thus, there is a
$\delta>0$ such that
\begin{equation}\label{eq3:100} |g(w)-g(x)|<2^{-m-n-1} \text{ for every }
w\in (x-\delta,x+\delta)
\end{equation}
if $n>0$.  If $n=0$, then by (\ref{eq3:afinal})
\begin{equation}\label{eq3:200} |g(w)-g(x)|<\tau \text{ for every } w\in
(-\delta+x,\delta+x).
\end{equation}
Since $x\in\Psi(n)\setminus\bigcup_{k\leq
n-1}\Psi(k)$ and
$\bigcup_{k\leq n-1}\Psi(k)$ is closed
Lemma~\ref{lem:3}(ii), implies that
\begin{equation}\label{eq3:101} f|_{\Psi(n)} \text{ is continuous at $x$.}
\end{equation}

We consider the case when $n=0$.
By Lemma~\ref{lem:3}(ii),
$f[\Psi(0)]=\{0\}$.  Since $\Theta_H(0)\prec_G\Psi(0)$, we may find a
bilateral sequence
$\{x_k\}_{k\in\omega}$ in
$\cont(g)\cap\Psi(0)$ such that $\lim_{k\to\infty} x_k=x$.   By (iii) of
Lemma~\ref{lem:3},
$\acc_{\cont(f)}(f,x_k)\supseteq(-\tau,+\tau)$.  Since
$\cont(f)\subseteq\cont(g)$ and
$x_k\in\cont(g)$ for each $k$, it follows from (c) of
Lemma~\ref{lem:suff} that
\begin{equation}\label{eq3:400}
\wacc_{\cont(f+g)}(f+g,x_k)\supseteq  [g(x_k)-\tau,g(x_k)+\tau]
\end{equation} for all $k\in\omega$.   If $\tau=\infty$, then, by
(\ref{eq3:400}), we have
$\wacc_{\cont(f+g)}(f+g,x)\supseteq (-\infty,+\infty)$; and so
$(f+g)(x)\in\wacc_{\cont(f+g)}(f+g,x)$.   If $\tau<\infty$, then we may
assume, taking a subsequence if necessary, that there is an $r\in\real$
such  that $\lim_{k\to\infty}(f+g)(x_k)=\lim_{k\to\infty}g(x_k)=r$.   So,
by (\ref{eq3:400}),
\[\wacc_{\cont(f+g)}(f+g,x)\supseteq [r-\tau,r+\tau].\]   By
(\ref{eq3:200}), $|(f+g)(x)-r|\leq\tau$.  Thus,
$(f+g)(x)\in\wacc_{\cont(f+g)}(f+g,x)$ when $n=0$.

We now consider the case when $n>0$.     Since
$\Theta_H(n)\prec_G\Psi(n)$, we may find a bilateral sequence
$\{x_k\}_{k\in\omega}$ in
$\cont(g)\cap\Psi(n)$ such that $\lim x_k=x$.  Taking a  subsequence if
necessary, we may assume, since $x\notin\Theta_H(0)$, that there is a number
$r$ such that $\lim_{k\to\infty} (f+g)(x_k)=r$.  By (iv) of  Lemma~\ref{lem:3},
\[\acc_{\cont(f)}(f,x_k)\supseteq(-2^{-n-m}+f(x_k),2^{-n-m}+f(x_k)).\]
Since $\cont(f)\subseteq\cont(g)$ and
$x_k\in\cont(g)$ for each $k$ it follows from (c) of
Lemma~\ref{lem:suff} that
\begin{equation}\label{eq3:500}
\wacc_{\cont(f+g)}(f+g,x_k)\supseteq
[-2^{-n-m}+(f+g)(x_k),2^{-n-m}+(f+g)(x_k)]
\end{equation}  for all $k\in\omega$. By (\ref{eq3:500}) and
$\lim_{k\to\infty}(f+g)(x_k)=r$, we have
\begin{equation}\label{eq3:102}
\wacc_{\cont(f+g)}(f+g,x)\supseteq  [-2^{-n-m}+r,2^{n-m}+r].
\end{equation} Now, by (\ref{eq3:100}) and (\ref{eq3:101}),
$\lim_{k\to\infty}f(x_k)=f(x)$ and $|g(x)-g(x_k)|<2^{-n-m-1}$ for all
$k$ large enough.  It follows that
\begin{equation}\label{eq3:103}
|(f+g)(x)-r|\leq|\lim_{k\to\infty}(f(x)-f(x_k)+g(x)-g(x_k))|\leq
2^{-n-m-1}.
\end{equation}
By (\ref{eq3:103}) and (\ref{eq3:102}), we have that
$(f+g)(x)\in\wacc_{\cont(f+g)}(f+g,x)$ when $n>0$.   Thus,
(\ref{eq3:nuff}) is established, and so
$f+g\in\quasi\cap\dar$.\qed

\bigskip

\noindent{\sc Proof of Theorem~\ref{thm:12}}.   Notice that in the proof
of Theorem~\ref{thm:1} the only place we needed to use the fact that
$f+H\subseteq\baire_1$ was when  we used (a) of Lemma~\ref{lem:suff} to
say that  (\ref{eq3:nuff}) implied $f+g\subseteq\dar\cap\quasi$.  So, if we
had just wanted to  prove that $f+H\subseteq\quasi\cap\pr$, we could have
just assumed that
$H\subseteq\cliq$ and used (b) of Lemma~\ref{lem:suff} to  say that
$f+g\in\quasi\cap\pr$.\qed









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