The important recent developments in set theoretical analysis concern the cardinal functions that are defined for different classes of real functions. These investigations seem to be analogous to those concerning of cardinal functions in topology from the 1970's and 1980's. (See [71, 67, 72, 139].) They are also related to the deep studies of cardinal invariants associated with different small subsets of the real line. (For a summary of the results concerning cardinals related to the measure and category see [57] or [7]. For a survey concerning cardinals associated with the thin sets derived from harmonic analysis see [17].)

The first group of functions is motivated by the notion of countable continuity
and was introduced in 1991 by J. Cichon, M. Morayne, J. Pawlikowski, and
S. Solecki in [21]. More precisely, they define the *
decomposition* function
for arbitrary families and
, where stands for the
set of all functions from *X* to *Y*.

where denotes the family of all coverings of with at
most many sets. In particular, if stands for the family of all
continuous functions (from subsets of into ) then

In [21] the authors considered the values of
for , where
stands for the functions of -th Baire class.

The motivation for this definition comes from a question of N. N. Luzin whether every Borel function is countable continuous. This question was answered negatively by P. S. Novikov (see [74]) and was subsequently generalized by Keldys [74] (in 1934), and S. I. Adian and P. S. Novikov [1] (in 1958). The most general result in this direction was obtained in late 1980's by M. Laczkovich [88], who proved, in particular, that for every .

One of the most interesting results from the paper [21] is the following theorem.

It has been also shown by J. Steprans and S. Shelah that none of these inequalities can be replaced by the equation.

There are also some interesting results concerning the value of , where is the class of all (partial) differentiable functions. It has been proved by Morayne [136, Thm 6.1,] that

Also, Steprans proved that

However, the relation between numbers , and for is unclear.

In the same direction, K. Ciesielski recently noticed that (obviously)

and that it is the best that can be said in ZFC.

In fact, (1) happens in a model obtained by extending a ground model with GCH by adding many Cohen reals. The equation follows immediately from Theorem 2.9.

The model for (2) is obtained as follows.
You start with a model with GCH,
assume that
and take an increasing sequence
cofinal with and such that each is a cardinal successor.
The desired model is obtained by a generic extension
via forcing *P* which a finite support iteration of forcings
, where each is a standard ccc forcing
adding the Martin's Axiom over the previous model and making
.

The second group of cardinal functions is defined in terms of algebraic operations on functions. Their definition was motivated by the following property of Darboux functions (from to ) due to Fast and mentioned in the previous section:

where |The extra assumption that is added in the definition of since otherwise for every family containing constant zero function we would have .

Its easy to see that the functions and are monotone in a sense
that
and for every
. Also clearly (1) is false for
. Thus, in language of the function Fast and
Kellum's results can be expressed as follows:

If (so, under the Generalized Continuum
Hypothesis GCH) the values of and are clear:
. Thus, Natkaniec
asked [104, p. 495,] (see also [63, Problem 1,]) whether the equation
can be proved in ZFC.

This question was investigated by Ciesielski and Miller in 1994. They proved that , that the cofinality of is greater than and that this, together with the inequalities is essentially all that can be proved in ZFC.

In particular, Theorem 4.7 says that
does not have to be a regular cardinal (part (d)) and that
can be any regular cardinal number between
and , with being ``arbitrarily large'' (part (c)).

At the same time Natkaniec and Rec aw established the values of and proving

The first systematic study of functions and was done by Ciesielski and Rec aw in the later part of 1995. They collected basic properties of operators and , which are stated below, and found the values of and for some other classes of functions.

In particular, (4) from Proposition 4.9 shows that every function is a difference of two functions from a class if and only if .

To state the other results from [41] recall the definitions the following
classes of functions, where *X* is an arbitrary topological space.

- of
*connectivity functions*, i.e., such that the graph of*f*restricted to*C*(that is ) is connected in for every connected subset*C*of*X*. - of
*extendable functions*, i.e., such that there exists a connectivity function with*f*(*x*)=*g*(*x*,0) for every . - of
*peripherally continuous functions*, i.e., such that for every and any pair and of open neighborhoods of*x*and*f*(*x*), respectively, there exists an open neighborhood*W*of*x*with and , where and stand for the closure and the boundary of*W*, respectively.

For the generalized continuity classes of functions (from into ) defined so far we have the following proper inclusions , marked by arrows . (See [13].)

Chart 1.

In particular, inclusions , monotonicity of and Theorem 4.7(a) imply that . Similarly, Theorem 4.8 implies that . The values of and for the remaining classes are as follows.

Notice also that
. Thus, by monotonicity of
and the above theorem we obtain the following corollary.

The values of functions and for the class has been
studied by Ciesielski and Natkaniec. First they noticed that
if the definition of from page is used
then trivially , since for any function
with for some
we have for every .
Thus, they modified the definition of to

where

(Note that is equal to
as defined on page .)
With this agreement in place they proved the following result.

However, the following problems remain open.

Another systematic study of operator was done by F. Jordan in 1996. In
his study, he examined the values of where
and classes are chosen from those discussed
above. Notice that has the following very nice interpretation:

where . To make this
study non-trivial Jordan notes first that the value of does not
determine the value of :

This paper [69] contains also the following results.

The importance of the extra assumptions in (4) and (5) of Theorem 4.15 is not clear. In particular, the following problem is still open.

Note also that (4) and (5) of Theorem 4.15, and Theorem 4.12 imply immediately the following corollary.

Finally, the following three classes of functions have been brought to this picture.

- of functions having the
*Cantor Intermediate Value Property*, i.e., such that for every and for each Cantor set*K*between*f*(*x*) and*f*(*y*) there is a Cantor set*C*between*x*and*y*such that ; -
of functions having the
*Strong Cantor Intermediate Value Property*, i.e., such that for every and for each Cantor set*K*between*f*(*x*) and*f*(*y*) there is a Cantor set*C*between*x*and*y*such that and is continuous; - of functions having the
*Weak Cantor Intermediate Value Property*, i.e., such that for every with*f*(*x*)<*f*(*y*) there is a Cantor set*C*between*x*and*y*such that .

Chart 2: ``Darboux like'' functions.

Clearly the above inclusions, monotonicity of and , and
Theorem 4.10 imply immediately:

The values of functions and for the class , and for
the classes formed by the intersections of with each of the remaining
classes mentioned above were not studied too carefully so far. However,
obviously implying

Also, it follows from Theorem 3.10 that

while also

A stronger version of this last inequality follows also from the following
recent theorem of K. Banaszewski and Natkaniec.

In particular,

and

This last inequality has been recently improved by F. Jordan, who proved the
following.

This theorem gives the
value of for many classes that can be obtained intersecting classes
from Chart 2 and .

Several other operators similar to and have also been studied.
Thus, in 1995 Natkaniec [107] introduced the following operators connected
to the composition of functions, where stands for the family of all
constant functions.

He proved also the following.

Similar functions have been also studied by Ciesielski and
Natkaniec [38]:

where () is the set of all
for which there exists such that
(, respectively). In fact, the class
has the following nice characterization:

In [38] the authors proved that

Also, in a recent short survey paper [109] Natkaniec evaluated the
values of operators , ,
and for the class of *almost continuous functions
in sense of Husain*, i.e., such that
for every non-empty open set
.

Some other cardinal operators connected with composition and concerning some kind of coding were also studied by Ciesielski and Rec aw [41], Ciesielski and Natkaniec [38], and Natkaniec [109].

Another variant of function is connected to the families of bounded
functions. To define it properly the following notation is necessary. For a
family let
stand for all uniformly bounded families
, and let be the class of all bounded
functions
. Then we define

In 1994 Maliszewski [94] proved that

so that . Moreover, he proved that
if all functions in
are measurable (have Baire property), then we can also assume that the
``universal summand'' bounded function has the same property. Similar results
were also proved for families of Borel measurable functions.

The values of for the other classes of functions from Chart 1 has been investigated by Ciesielski and Maliszewski [36]. In particular, they proved

Notice also that Theorem 4.22 implies immediately
the following corollary.

In particular, Corollary 4.23(1) generalizes a
result of Darji and Humke [49] that every bounded function can be
expressed a sum of three bounded almost continuous functions. On the other
hand Corollary 4.23(2) shows that the following result of Natkaniec
is sharp.

It might be also interesting to examine a bounded version of , defined as

However this function has not been studied so far.

One might also consider the study of the operator (and ) for the
functions from into with *n*>1. This has indeed been done by
Ciesielski and Wojciechowski in [44]. The study concerned only the classes
, , , , and since
other classes from Chart 2 do not have natural generalizations into functions of
more than one variable. First, one should recall that for *n*>1 Chart 1 is not
valid any more. The new inclusions (for *n*>1) are as follows:

(The inclusion ``'' was proved by
Hamilton [66] and by Stallings [132], and the inclusion
``'' by Hagan [65]. The proof of the
inclusion ``''
is presented in [132]. The
examples showing that
and
can be found
in [105, Examples 1.1.9 and 1.1.10,] or
[104, Examples 1.7 and 1.6,],
while a simple Baire class 1 function in
was described in
[116, Example 1,].) We do not know whether the inclusion
is proper.

The problem with studying the value of the operator for all these
classes (except ) is that there exists a function
which is not a sum of *n* Darboux functions, implying that

However, every function function is sum of *n*+1 extendable
functions. To express these results nicely, define for
the *repeatability* of as the smallest integer *k* such that any function
can be expressed as a sum of *k* functions from
. (We put if such a number does not
exist.) In this language the results of Ciesielski and Wojciechowski can be
stated as follows.

Clearly Theorem 4.25 implies that .
The problem (stated in [44]) whether this equation can be replaced by the
equality has been recently solved by F. Jordan.

The value of is clearly equal to 2, since Natkaniec [104] proved that . This fact has been recently improved by F. Jordan, who proved

Notice also, that in the language of operator the results from
Theorem 4.24 and Corollary 4.23(2) can be expressed by the
equation

where is the natural generalization of for the
class of bounded functions.