A hierarchy in the family of real surjective functions

Abstract This expository paper focuses on the study of extreme surjective functions in ℝℝ. We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in ℝℝ.


Introduction
At the beginning of the 20th century Lebesgue [1] proved the existence of a mapping f W OE0; 1 ! OE0; 1 such that f .I / D OE0; 1 for every non-degenerate subinterval I of OE0; 1. Lebesgue's example can be adapted to construct a mapping defined on the whole real line that transforms every non-degenerate interval into R. This exotic property turns out to be shared by a surprisingly large class of functions that we call everywhere surjective.
We point out in Section 2 that everywhere surjective functions attain every real value at least @ 0 many times in every non-degenerate interval. In fact, it is possible to define an everywhere surjective function that attains each real number c many times in every non-degenerate interval, where c stands for the cardinality of R. An example of a function enjoying this refined form of extreme surjectivity will also be given. This example, far from being an isolated case, is just an instance of a very large class of functions called strongly everywhere surjective. The notion of strongly everywhere surjectivity does not exhaust all possibilities in the search of extreme surjectivity. Indeed, there are surjective functions satisfying even more restrictive conditions. We also construct a function that attains every real number c many times in every perfect set, which is obviously a much stronger form of surjectivity. These functions are called perfectly everywhere surjective. We can take an even further step forward towards "supreme surjectivity". In 1942, F. B. Jones [2] constructed a function whose graph intersects every closed set in R 2 with uncountable projection on the abscissa axis. The functions that satisfy this latter property are called Jones functions. It is easily seen that a Jones function is perfectly everywhere surjective. The class of Jones functions can be proved to be large from an algebraic point of view too.
In order to formalize what is meant by an "algebraically large set" the notion of lineability is commonly used. We say that a subset M of a linear space E is -lineable, if M [ f0g contains a linear subspace of E of dimension . If M [ f0g contains an infinite dimensional linear space we simply say that M is lineable. In Section 3 we will show, among other important results, that the four classes of surjective functions mentioned above are lineable, actually 2 c -lineable. It is important to mention that the study of the lineability of sets of strange functions has become a fruitful field since the term "lineability" was coined in 2005 (see [3]). A thorough description of the most relevant lineability problems and other related topics can be found in the monograph [4] or in the expository paper [5]. The interested reader may also consult the references [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].
In Section 4 we establish the connection existing between the classes of extremely surjective functions defined in Section 2 with other classes of interesting functions. We consider the sets of additive (that is, Q-linear) mappings, functions with a dense graph, Darboux functions and Sierpiński-Zygmund functions.
This survey paper is written in such a way that it is accessible to the largest possible audience. For this reason we provide a good account of examples, which are presented in detailed for completeness. We also give full proofs of most of the lineability problems introduced in Section 3. We have included the proofs of several well-known topological results in order to make the paper as inclusive and self-contained as possible. However we have decided to omit the proofs that either are too complex or require complicated techniques of set theory.
We will use the following standard definitions and notations: R R stands for the set of all mappings from R to R. D denotes the subset of R R of the Darboux functions, i.e., functions that transform intervals into intervals. S, C and I will denote, respectively, the sets of surjective, continuous and injective functions from R to R. If C R 2 , then dom.C / denotes the projection of C on the abscissa axis. If f 2 R R , we will often denote the graph of f , graph.f / WD f.x; f .x// W x 2 Rg, simply by f .

A few examples of extreme surjective functions
In this section we provide a few examples of surjective functions enjoying the property that they transform every non-degenerate interval into the whole real line. We will see that there is a hierarchy among the functions satisfying this property. Let us see first several examples of everywhere surjective functions.

Everywhere surjective functions
First recall that a mapping f W R ! R is everywhere surjective if it transforms non-degenerate intervals into the whole real line, or equivalently, if f ..a; b// D R, for all a; b 2 R with a < b. The set of all everywhere surjective mappings is represented by ES. The construction of one everywhere surjective function is not trivial. The first known example of such a function dates back to Lebesgue and is more than a century old. Here we give several more modern examples. The first of them appears in [24] and is presented below in detail for the sake of completeness.
otherwise, then f satisfies the following properties: 3. f is surjective on every non-degenerate interval.
In order to prove the assertions 1-3 the following remark can be useful: br.n C 1/c n C 1 D r; for each r 2 R. Indeed, x 1 Ä bxc Ä x, 8x 2 R. If we set in the previous inequalities x D r.n C 1/ for arbitrary r 2 R and n 2 N, then r.n C 1/ 1 Ä br.n C 1/c Ä r.n C 1/. Dividing by n C 1 we arrive at Finally, taking limits we conclude that lim n!1 br.n C 1/c n C 1 D r: Proof.
(1) Given x 2 R and q 2 Q , 9r; s 2 Z such that q D r s . If n s, we have that nŠq D nŠ r s 2 Z. Thus nŠ x nŠ .x C q/ is a multiple of . Therefore tan.nŠ .x C q// D tan.nŠ x/, 8n s. If the limit does not exist, by definition we have (2) Given y 2 R we choose r 2 OE0; 1/ such that tan. r/ D y. Let x 2 R be given by Of course x D x n C n . Notice that nŠx n 2 Z, 8n, and hence, by the previous step we have that tan.nŠ x/ D tan.nŠ n /, 8n. Therefore tan.nŠ n / D tan. r/ D y: (3) Assume that a; b; y 2 R with a < b. By (2) there exists u 2 R such that f .u/ D y, and by (1) we have that Observe that tan.nŠ q/ ! 0 as n ! 1. We conclude that ff .x/ W a < x < bg D R.
The second construction we present in this section is based on the fact that every interval contains a Cantor like set and that Cantor sets are uncountable. The example is taken from [23] (see also [4], [5] and [15]).
We construct a mapping f W R ! R as follows: Let I n n2N be a sequence containing all the intervals with rational endpoints. Then I 1 contains a Cantor like set, which we denote by C 1 . On the other hand, I 2 n C 1 , contains another Cantor like set, which is denoted by C 2 . Now the set I 3 n .C 1 [ C 2 / contains a new Cantor like set, namely C 3 . Repeating this process, we construct by induction a sequence C n n2N of pairwise disjoint Cantor like sets, such that I n n S n 1 kD1 C k C n . Since C n is uncountable, there exists a bijection ' n W C n ! R, for every n 2 N. It is now that we define f W R ! R by Finally, if I R is a non-degenerate interval, then there exists k 2 N with I I k . By construction of C n we have that I k C k and hence, by definition of f , f .I / f .I k / f .C k / D ' k .C k / D R: Interestingly, the mapping f is null almost everywhere in R.
The third example we provide is based on the fact that there is a partition of R into c many dense sets. This can be achieved by considering the relationship in R given by x y , x y 2 Q .x; y 2 R/: The equivalence classes have the form OE˛ D˛C Q and are obviously pairwise disjoint, dense sets in R. Since OE˛ is countable and R D S˛2 R OE˛, it is obvious that R= contains c elements.
for every n 2 N. Since f 2 ES, there exists x n 2 I n I such that f .x n / D y. Since the I n 's are pairwise-disjoint, we have constructed a sequence .x n / of distinct points in .a; b/ such that f .x n / D y.
The next lemma will be very useful throughout the paper. For instance, if we apply it to the family f.a; b/ fyg W a; b; y 2 R and a < bg; we obtain again an example of a function in ES. We recall that if A R 2 then dom.A/ denotes the projection of A over the abscissa axis.
Lemma 2.6. Let fA˛g˛< c be a family of subsets in R 2 such that card.dom.A˛// D c, for each˛< c. Then there exists a function f 2 R R such that f \ A˛¤ ¿.
Among the functions in ES there are some that are yet more surjective since they are able to attain every real number uncountably many times in each non-degenerate interval of R. We introduce these functions in the next subsection.

Strongly everywhere surjective functions
Recall that f W R ! R is strongly everywhere surjective if f attains every real number c many times in each nondegenerate interval of R. The set of all the strongly everywhere surjective functions is denoted by SES. Obviously, we have that SES ES. Let us check first that SES is non-empty.
Recall that the Cantor set is homeomorphic to f0; 1g N , which, in its turn is homeomorphic to Since the f0; 1g N f˛g's are pairwise disjoint sets homeomrphic to the Cantor set, we have the following: An example of a mapping in SES can be constructed using Lemma 2.7 by adapting Example 2.3. The example is taken from [15] (see also [4] and [5]).
Example 2.8. In Example 2.3 we had a sequence C n n2N of pairwise disjoint, Cantor like sets such that I n n S n 1 kD1 C k C n for every n 2 N. Now, according to Lemma 2.7, for each n 2 N there is a partition fC i n W i 2 Rg of C n consisting of Cantor-like sets. Since the C i n 's are uncountable, for each n 2 R and i 2 R there exists a bijection It only remains to show that f is strongly everywhere surjective. Indeed, take I R a non-degenerate interval. Then there exists k 2 N with I I k . For this k we have f .
Also, f attains obviously every real number c times in I . Remark 2.9. Notice that the function constructed in Example 2.4 attains every real number only countably many times in every interval, and therefore it is ES but not SES. Hence

SES¨ES :
In the next section we will see that in the case where the Continuum Hypothesis is not assumed, there is a hierarchy of degrees of surjectivity between the classes ES and SES.

Everywhere Ä-surjective functions
If Ä is a cardinal number such that @ 0 Ä Ä Ä c, we say that a function f 2 R R is everywhere Ä-surjective if for every y 2 R, f attains y at least Ä times in every non-degenerate interval. We denote by ES Ä the set consisting of all the everywhere Ä-surjective functions in R R .
Observe that - Given Ä such that @ 0 Ä Ä < c, we construct in the following example an everywhere Ä-surjective function that is not everywhere Ä C -surjective. This shows that Example 2.10. Let Ä be a cardinal number with @ 0 Ä Ä Ä c and consider fD˛W˛2 Rg the partition of R into c many dense, countable sets constructed in the comments preceding Example 2.4. Now consider a partition fÄˇWˇ2 Rg of R into c many sets Äˇof cardinality Ä and set Also, observe that f attains every real number exactly Ä times in every non-degenerate interval, which shows, additionally, that f cannot be everywhere -surjective for every > Ä.
The functions in SES might seem sufficiently special or pathological, however it is possible to construct even more surprising functions in the class SES, as we will see in the next two sections.

Perfectly everywhere surjective functions
Observe that in the definition of strong everywhere surjectivity we can restrict ourselves without loss of generality to closed, non-degenerate intervals. In other words, a function f W R ! R is strongly everywhere surjective if and only if f attains every real number c times in every non-degenerate, closed interval. Now, a non-degenerate, closed interval is a simple example of perfect set. We recall that P R is perfect if P 0 D P . The question that arises now is whether a strongly everywhere surjective function attains each real number c times in every perfect set. The answer to this question is no. Indeed, we just need to consider the function defined in Example 2.8, which is SES. However f attains every real number only once in each Cantor set C i n , which is perfect. From now on, we will say that f W R ! R is perfectly everywhere surjective if f is surjective on every perfect set. The set of all perfectly everywhere surjective mappings is denoted by PES. We will see later that f 2 PES if and only if f attains every real number c times in every perfect set P R. This shows that the elements of PES represent a stronger form of surjectivity than the elements of SES. The example of a PES function we provide here is taken from [15]. In its construction we will need the following well-known fact, whose proof is given for completeness.
Lemma 2.11. If P is perfect, then card.P / D c.
Proof. Without loss of generality, we can assume that P is bounded. Then, since P is closed, there exist˛D min P andˇD max P . Then P Â OE˛;ˇ. If m is the middle point of OE˛;ˇ, we define There is an even stronger form of surjectivity than perfectly everywhere surjective functions that will be studied in the next section.

Jones functions
In 1942, F. B. Jones [2] found an example of a function in R R such that for any closed subset C R 2 with uncountable projection over the abscissa axis, f \ C ¤ ¿. A function satisfying this property is called a Jones function. The set of all Jones functions is denoted by J . (Notice that, since dom.C / is -compact, then uncountable is equivalent to cardinality c in the previous definition.) Example 2.14. In order to obtain a function f 2 J, we just need to apply Lemma 2.6 to the family fC R 2 W C is closed and card.dom.C // D cg: Observe that if f 2 J, then f 2 PES since P fyg is closed in R 2 for all perfect set P R. Therefore J PES.
Consider the function f constructed in the proof of Lemma 2.6 for the family f.P n fyg/ fyg W P R is perfect and y 2 Rg : Then f \ C D ¿ where C is the closed set f.x; x/ W x 1g. Hence f 2 PES but f … J, and therefore J¨PES :

Algebraic size of sets of surjective functions
In this section we discuss the algebraic size of the sets ES, ES Ä , SES, PES and J from the lineability viewpoint.
In order to prove that ES, SES and PES are 2 c -lineable, the following result will be crucial. We reproduce the original proof for completeness: Lemma 3.1 (Aron et al. [3]). There exists a vector subspace V 0 of R R whose dimension is 2 c such that every nonnull element of V 0 is surjective. In other words, S is 2 c -lineable.
Proof. Let ' W R ! R N be a bijection that transforms .0; 1/ into the set of sequences whose first element is 0. For each A R, we define Where I A is the characteristic function of A. We have the following: (a) The family fH A W A R; A ¤ ¿g is linearly independent. In order to prove it, let us consider m different subsets C 1 ; C 2 ; ; C m , of R and m non-null numbers 1 ; 2 ; : : : ; m . Assume that Since the C j 's are different, there exists k 2 f1; 2; : : : ; mg and x j such that x j 2 C k n C j for each j ¤ k. In order to see the latter, assume that for every k 2 f1; : : : ; mg, there exists j ¤ k such that C k n C j D ¿. This would be equivalent to saying that for all k 2 f1; : : : ; mg there exists j ¤ k with C k C j . Renaming the sets if necessary, we would have: C 1 C 2 : : : C m C˛; where˛2 f1; :::; m 1g. This would imply that at least two sets coincide, which is a contradiction. Now, we can set, without loss of generality, that k D m. let x D .1; x 1 ; x 2 ; : : : ; x m 2 ; x m 1 ; x m 1 ; x m 1 ; : : : /: Remark 3.2. In connection with Lemma 3.1, the reader may find of interest the fact that S \ C is c-lineable. To see this we just need to realize that the span of fe rx e rx W r 2 .0; 1/g is a c-dimensional space contained in .S \ C/ [ f0g.  It turns out that J is 2 c -lineable too. This is proved in Theorem 3.6 below. Since J is a subset of all the other classes of surjective functions introduced in Section 2, Theorem 3.6 also proves that ES, SES and PES are 2 c -lineable. From this viewpoint Theorem 3.3 (and hence Lemma 3.1 too) would be unnecessary. We have decided to include Theorem 3.3 because its proof is accessible to a much larger audience.
The proof of the 2 c -lineability is based on a couple of topological results about Bernstein sets. We recall that B Â R is a Bernstein set if for every perfect set P R, we have that B \ P ¤ ¿ and .R n B/ \ P ¤ ¿. Proof. It suffices to find in R c many pairwise disjoint sets in R, B˛,˛< c, such that B˛is perfectly dense, i.e., B˛\ P ¤ ¿ for every perfect set P R. Indeed, if˛<ˇ< c we have also Bˇ\ P ¤ ¿, so .R n B˛/ \ P ¤ ¿, and hence B˛is a Bernstein set.
In principle there is no need to assume that [ <c B˛D R: In order to see the latter, suppose we have already constructed a family fB˛W˛< cg of pairwise disjoint Bernstein sets and enumerate where Ä Ä c. If we set then these new sets are also pairwise disjoint, Bersntein sets and their union is R. Let us enumerate the perfect sets of R as fPˇWˇ< cg. We just need to construct by transfinite induction a double sequence .x˛ˇ/˛;ˇ< c of different elements in R in such a way that x˛ˇ2 Pˇfor all˛;ˇ< c because in that case the sets B˛D fx˛ˇWˇ< cg satisfy what we need.
Suppose that in the step of the induction we have constructed the elements x˛ˇ, where˛;ˇ< . Since the cardinality of the constructed elements is 2 < c, we can choose 2 C 1 additional elements, namely, x˛ 2 P with˛< and x ˇ2 PˇwithˇÄ . Therefore we have constructed fx˛ˇW˛;ˇÄ g. Lemma 3.5. Let B be a Bernstein set. There exists a Jones function f such that for all g 2 R R such that f j B Á gj B then g is a Jones function.
Proof. It is enough to apply Lemma 2.6 to the family f.B R/ \ C W C R 2 is closed and card.dom.C // D cg: To show that this family satisfies the hipothesis of Lemma 2.6 it suffices to prove that card.B \dom.C // D c because dom..B R/ \ C / D B \ dom.C /. Indeed, dom.C / is a -compact set of cardinality c and therefore at least one of the compact sets that form the union must have cardinality c. Hence that compact is the union of a perfect set and a countable set (see [25]) and so it must contain a perfect set. On the other hand, any perfect set contains a Cantor-like set C . Taking into account Lemma 2.7 it is straightforward that card.B \ C / D c.
Proof. Let fB˛W˛< cg as in Lemma 3.4. For each˛< c let f˛be a function in J such that every g 2 R R with f˛j B˛Á gj B˛s atisfies that g 2 J (see Lemma 3.5). We can also assume that f˛j RnB˛Á 0. Now consider the set Observe that V is clearly a linear space and that every non-null element of V is in J by Lemma 3.5 because if '.ˇ/ ¤ 0 for someˇ< c then P˛< c '.˛/f˛coincides with fˇin Bˇ. Also, V is isomorphic to R c , whose cardinality is 2 c , which concludes the proof. Remark 3.7. As mentioned above it is interesting to observe that the space defined in Theorem 3.6 also proves that the other classes of surjective functions introduced in Section 2, namely ES, ES Ä , SES and PES are also 2 c -lineable since J is a subset of them.
Another fact that reveals that the size of J (and hence the size of ES, ES Ä , SES and PES too) is enormous, is shown by the following result, whose proof can be deduced from the fact that the additivity of J is bigger than 2 (see [16] for details). However, we give below our own proof: Proof. For f 2 R R , let us consider the family W C is closed and card.dom.C // D cg; F 2 W D˚f.x; y C f .x// W .x; y/ 2 C g W C R 2 is closed and card.dom.C // D c « : Let g 2 R R be the function constructed in Lemma 2.6 for the family F. Since g \ C ¤ ¿ for all closed C R 2 with card.dom.C // D c we have that g 2 J. We also have that .g f / \ C ¤ ¿ for all closed C R 2 with card.dom.C // D c, which implies that g f 2 J, and hence h WD f g 2 J.
In the rest of this section we present a series of results showing what is known nowadays about the algebraic size of the sets S n ES, ES n ES Ä with @ 0 Ä < Ä Ä c, SES n PES and PES n J. Among the above problems we know the optimal solution to only two of them: For the rest of the cases we only have partial and probably not optimal answers. Theorem 3.10 (Bartoszewicz et al. [8,Theorem 3.12]). If @ 0 Ä < Ä Ä c then ES n ES Ä is 2 -lineable. Remark 3.11. Although it is not explicitly shown in [8], it can be deduced that ES n ES Ä is 2 -lineable for every < Ä since, in that case, ES n ES Ä ES n ES Ä and, by Theorem 3.10 is 2 -lineable. The case D @ 0 and Ä D c is explicitly studied in [11,Theorem 2.14]. Also, it can be proved that the result given in Theorem 3.10 is not optimal in general. If we admit Martin's Axiom, 2 D c for all @ 0 Ä < c. However we have the following result: Theorem 3.12 (Ciesielski et al. [11,Corollary 2.15]). The set ES n SES is c C -lineable.
The estimate given in Theorem 3.12 implies that ES n SES is 2 c -lineable under CH (Continuum Hypothesis). Whether or not ES n SES is 2 c -lineable in ZFC (Zermelo-Fraenkel Theory with Axiom of Choice) is still an open question.
We do not know much about the size of the set PES n J. We do not even know whether this family is lineable or not.

Relationship between extremely surjective functions and other classes
In this section we will study the relationship between the class ES and other families of interesting functions like, for instance, the class of additive mappings (or equivalently, Q-linear), the class of function in R R with dense graph in R 2 , the class of Darboux functions and the set of Sierpińiski-Zygmund functions. We will deal in the first place with additive mappings and functions with a dense graph.

Everywhere surjective functions, additive mappings and functions with a dense graph
Recall that f 2 R R is addtive if f .x C y/ D f .x/ C f .y/ for all x; y 2 R. It is easy to prove that a function is Q-linear if and only if it is additive. We denote the sets of additive mappings and the set of functions with a dense graph, respectively by Add and DG. The classes DG and Add are related to ES as follows: (a) ES DG, which is obvious, and (b) ES \ Add D Add \.S nI/. Recall that S and I denote, respectively, the surjective and injective elements of R R .
In order to see (b), we reproduce the argument used in [20]. Observe first that f W R ! R is in ES if and only if f 1 .t / is dense for all t 2 R. Also, any 1-dimensional Q-subspace of R is dense, and therefore any proper Q-subspace of R is dense too. Since ES \ Add Add \.S nI/ is trivially true, assume that f 2 Add \.S nI/. Since f is surjective, for every t 2 R there exists x 2 R with f .x/ D t . Notice that f 1 .t/ D x C ker.f /. Also ker.f / is dense. Indeed, since f is not injective, ker .f / ¤ f0g, and hence the Q-subspace f 1 .0/ D ker .f / is dense. We conclude that f 1 .t / is dense for all t, or in other words, f 2 ES \ Add.
It is easy to prove that ES \ Add ¤ ¿. Indeed, if H D fh i W i < cg is a Hamel basis, we just need to define f on H such that f is surjective and not injective. Extending f to R by linearity we obtain an additive mapping in ES \ Add. In fact we have a much stronger result whose original proof, for completeness, is given below: Theorem 4.1 (García-Pacheco et al. [20]). The set ES \ Add is 2 c -lineable.
Proof. Consider a Hamel basis I of R regarded as a Q-linear space and letˆW I ! R be bijective. Define where V 0 is a 2 c -dimensional space of surjective functions (except for the zero function). Clearly card.W / D 2 c and each non-null element f W I ! R of W is a surjective function that can be extended by linearity, uniquely, to It is interesting to observe that the non-null elements of the space V 0 introduced in Lemma 3.1 are not Q-linear. Indeed, using the terminology of the proof of Lemma 3.1, for each f 2 V 0 and x 2 .0; 1/, we have f .x/ D .H A ı '/ .x/ D H A .0; x 1 ; x 2 ; :::/ D 0 Q 1 i D1 I A .x i / D 0 from which f is neither injective nor lies in ES. Hence f cannot be Q-linear. It is still possible to prove that ES n Add is not only empty, but also algebraically large. We give the proof for completeness.
Proof. Choose f 2 ES \ Add and define W D fg ı f W g 2 V 0 g with V 0 as in Lemma 3.1. It is easily seen that W is a 2 c -dimensional space (isomorphic to V 0 ) whose non-null elements are in ES.
On the other hand, if g 2 V 0 n f0g, since g is not additive, there exist x; y 2 R such that g .x C y/ ¤ g .x/ C g .y/ : Then which shows that g ı f is not additive. Hence W .ES n Add/ [ f0g and the proof is finished.
Next we study the lineability of the set DG \ Add n ES. We reproduce the original proof for completeness.  [20]). The set DG \ Add n ES is 2 c -lineable.
Proof. Let I be a Hamel basis of R regarded as a Q-linear space. Fix i 2 I and consider a bijection W I ! I n fi g.
It is straightforward to prove that can be extended by linearity to an injective Q-linear mappingˆW R ! R.
It can also be proved thatˆ.R/ D R. Indeed, choose > 0 and p 2 R nˆ.R/, and consider j 2 I n fi g and 2 Q such that j˛j pj < . Since there is s 2 I with .s/ D j , we have that jˆ.˛s/ pj D j˛ .s/ pj D j˛j pj < .
Define now U D fˆı g W g 2 W g; where W is any 2 c -dimensional linear space such that W Â .ES \ Add/ [ f0g (see Theorem 4.1). It is clear that U is a 2 c -dimensional linear space and that every non-null element of U is Q-linear and not surjective. Also, f maps every non-degenerate interval toˆ.R/, which completes the proof.
To finish this section we have included a result on the elements of Add n DG. Proof. Choose f 2 Add. If f 2 DG, then f is obviously discontinuous. If we assume now that f is not continuous, then f cannot be homogeneous, and hence there does not exist c 2 R such that f .x/ D cx for all x 2 R. If we take The vectors v 1 D .x 1 ; f .x 1 // and v 2 D .x 2 ; f .x 2 // are clearly linearly independent and therefore they generate R. If q 1 ; q 2 2 Q, we can approximate q 1 v 1 C q 2 v 2 to any vector v since Q is dense in R 2 . Therefore In other words f 2 DG.
The next section is devoted to the study of the linear structure of the set of the Daboux functions in R R .

Darboux functions
We recall that f 2 R R is Darboux if it transforms intervals into intervals and that D represents the set of Darboux functions. Obviously ES D and therefore D is 2 c -lineable. We also have the following interesting results: If we consider the 2 c -dimensional linear space W D ff ] W f 2 V g, then it is plain that W .D n ES/ [ f0g.
Theorem 4.6. The set S n D is 2 c -lineable.
Proof. Observe that the space V generated by the characteristic functions of subsets of . 1; 0 has cardinality 2 c and hence it is 2 c -dimensional. Let B 1 D fe˛W˛< 2 c g be a basis for V . Now let us consider a basis B 2 D ff˛W < 2 c g of V 0 , where V 0 is as in Lemma 3.1. If for each˛< 2 c we define then the span of fg˛W˛< 2 c g is a 2 c -dimensional space contained in .S n D/ [ f0g.

Sierpiński-Zygmund functions
The construction of a Sierpiński-Zygmund function is motivated by the following result: Theorem 4.7 (Blumberg [26]). For every f 2 R R there exists a dense set Z R such that f j Z is continuous.
The set Z provided in Blumberg's proof turns out to be countable. Sierpiński and Zygmund asked whether or not an uncountable set could be found satisfying Theorem 4.7. This led them in 1923 ( [27]; see also [28, pp. 165,166]) to the construction of an instance of what nowadays it is known as a Sierpiński-Zygmund function. We recall that f 2 R R is Sierpiński-Zygmund if for every Z R with cardinality c, the restriction f j Z is not continuous. We denote the set of Sierpiński-Zygmund functions by SZ.
If CH holds, the restriction of a Sierpiński-Zygmund function to any uncountable set cannot be continuous. The Continuum Hypothesis is necessary in this setting. Shinoda proved in [29] that if Martin's Axiom and the negation of CH hold, and @ 0 < Ä < c then for every f 2 R R there exists a set Z R of cardinality Ä such that f j Z is continuous.
It is interesting to observe that Sierpiński-Zygmund's example satisfies a stronger condition, namely f j Z is not Borel for all Z R with card.Z/ D c, which is a stronger condition than that of the definition of Sierpiński-Zygmund function. This motivates the following definition SZ.Bor/ WD˚f 2 R R W 8Z R with cardinality c, the restriction f j Z is not Borel « : Obviously SZ.Bor/ SZ. The question is whether or not SZ.Bor/ D SZ is undecidable under the usual set theoretic settings. However, if dec.Bor; C/ denotes the minimal cardinal Ä such that for every Borel function f W X ! R there is a partition .X˛/˛< Ä of X with f j X˛c ontinuous for all˛< Ä, then it can be proved that: Another interesting question to be considered is that the standard axioms of set theory (like ZFC) do not guarantee the existence of Sierpiński-Zygmund functions that are surjective or Darboux. However, the following can be proved assuming stronger hypothesis: Theorem 4.9 (Ciesielski et al., [11]). If cov.M/ D c, i.e., the union of less than continuum many meager sets does not cover R, then SZ \ ES is c C -lineable.
Observe that Martin's Axiom (see [30]) implies the condition cov.M/ D c. However, assuming different set of theoretic hypotheses, it is possible to prove that SZ and ES are even disjoint: Theorem 4.10 (Balcernak et al., [31]). Under the CPA, Covering Property Axiom (see [32] for details), we have that SZ \.D [ S/ D ¿ (hence SZ \ ES D ¿).

Conclusions and open questions
The diagram in Figure 1 shows how some of the classes introduced in this paper are related to each other.  1. We know that ES n SES is c C -lineable (see Theorem 3.12). However, we do not know whether c C is optimal or not. 2. The optimal lineability of ES n ES Ä with @ 0 Ä < Ä Ä c is not known. 3. Nothing is known about the lineability of the set PES n J.