Structural aspects of the non-uniformly continuous functions and the unbounded functions within C(X)

STRUCTURAL ASPECTS OF THE NON-UNIFORMLY CONTINUOUS FUNCTIONS AND THE UNBOUNDED FUNCTIONS WITHIN C(X) RAFAEL AYALA-G´ OMEZ, LUIS BERNAL-GONZ´ ALEZ, MAR´ IA DEL CARMEN CALDER´ ON-MORENO, AND JOS´ E ANTONIO VILCHES-ALARC ´ ON Dedicated to the loving memory of our colleague Bernardo Cascales (1958-2018) Abstract. We prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then, under appropriate mild assumptions, there exists in fact a plethora of such functions, in both topological and algebraical senses. Corresponding results are also obtained concerning unbounded continuous functions on a non-compact metrizable space. 1. Introduction It is a well known fact that, if (X, d) and (Y, ρ) are metric spaces such that (X, d) is compact, then any continuous mapping f:X→Y is uniformly continuous, that is, it satisfies that, given ε > 0, there is δ > 0 such that ρ(f(x), f(x0)) < ε for every pair of points x, x0∈Xwith d(x, x0)< δ (see e.g. [40, Chap. 3]). However, the reciprocal assertion is not true, even though we consider (Y, ρ)=(R, de), the real line endowed with the Euclidean distance. Indeed, if Xis an infinite set and we endow it with the discrete metric ρd(x, y) = 0 if x=y 1 if x6=y then (X, d) is not compact but any function (in particular, any continuous function) X→Ris uniformly continuous. Recall that uniform continuity depends strongly upon the metrics. For instance, if X={1/n : n= 1,2, . . . }and f(1/n) := n(n≥1), then deand ρdare equivalent metrics on X, but f: (X, ρd)→(R, de) is uniformly continuous while f: (X, de)→(R, de) is not. Turning to compact spaces, what is true is the following characterization –due to Hewitt [23] in 1948– of this kind of spaces: a metrizable space Xis compact if and only if every continuous function X→Ris bounded. 2010 Mathematics Subject Classification. 46E10, 15A03, 08B20, 54C05, 54E35. Key words and phrases. Atsuji space, non-uniformly continuous function, lineability, spaceability, algebrability. 1 2 AYALA, BERNAL, CALDER ´ ON, AND VILCHES In 1958 Atsuji [5] gave a fairly comprehensive list of characteristic properties of those metric spaces for which any real continuous function defined on them is uniformly continuous. From now on, these spaces will be called Atsuji spaces (they are also frequently called UC spaces in the literature). Specifically, and under this terminology, it was proved in [5] the following result, among others. As usual, if Xis a topological space and A⊂X, then we denote by A0the set of limit points of Ain X. We warn that along this paper the symbol ⊂does not necessarily mean strict inclusion. Theorem 1.1. Let (X, d)be a metric space. The following properties are equivalent: (a) (X, d)is an Atsuji space. (b) The subset X0is compact and, for each ε > 0, there exists δ > 0such that d(x1, x2)≥δwhenever x1and x2are distinct points satisfying d(x1, x)≥ε≤d(x2, x)for all x∈X0. Prior to this, spaces on which continuity equals uniform continuity had been studied by Nagata [41], Monteiro and Peixoto [37], and Isiwata [27]. Other characterizations of Atsuji spaces were provided by Rainwater [43], Hadziivanov [22], Mr´owka [39], Waterhouse [46], Wong [47], Toader [45], Hueber [25], Beer [9–11], Chaves [18], Bors´ık [16], and Jain and Kundu [28] (see also [29] for characterizations of those metric spaces whose completions are Atsuji spaces). Moreover, in 1955 Levine [33] was able to give necessary and sufficient conditions for a subset of the Euclidean real line R to be an Atsuji space, and this characterization was extended in 1960 by Levine and Saunders [34] to subsets of any prescribed metric space (X, d) (corresponding results when the final space is not necessarily (R, de) are also obtained in [34]). The problem of describing those metrizable topological spaces admitting a compatible Atsuji distance has been considered by Nagata [41], Levsenko [35], Rainwater [43], Mr´owka [39] and Beer [11]. It is important to mention that any Atsuji distance is complete (see [45]) and, since a metrizable space for which every compatible distance is complete is already compact (Niemytzki- Tychonoff’s theorem), one derives that each non-compact metrizable space has a compatible non-Atsuji distance. Moreover, characterizations of compactness of metric spaces in terms of uniform continuity have been provided by Hueber [25] and Snipes [44]. Starting from a non-Atsuji metric space (X, d), a natural question is the analysis of the size and structure of the family of continuous functions on Xthat are not uniformly continuous. Such analysis, which will be carried out in Section 3, is the main aim of this paper. It will be proved that, under appropriate conditions, the mentioned family is very large in both topological and algebraic senses, so as to contain large –that is, dense or closed infinite dimensional– vector subspaces as well as infinitely generated algebras. Corresponding results for size and structure of the family of unbounded continuous functions defined on a non-compact metrizable topological space NON-UNIFORMLY CONTINUOUS FUNCTIONS AND UNBOUNDED FUNCTIONS WITHIN C(X)3 will be provided in Section 4. Section 2 is devoted to exhibit the necessary notation and background in order to deal with the above mentioned question. 2. Notation, lineability and preliminary results Unless explicitly stated, the real line Rwill always be endowed with the Euclidean metric de(x, y) = |x−y|. As usual, Nwill denote the set of positive integers, while N0:= N∪{0}. We set c:= card (R), the cardinality of the continuum. If (X, d) is a metric space, x∈Xand ε > 0, then Bd(x, ε) represents the open ball {y∈X:d(x, y)< ε}. If Xand Yare topological spaces and A⊂Xthen Aand C(X, Y ) will stand, respectively, for the closure of Ain Xand for the set of continuous functions X→Y. Also, we define C(X) := C(X, R), which is a vector space. If Xand Yare metric spaces, then by UC(X, Y ) we represent the family of all uniformly continuous functions X→Y. Consistently, we denote UC(X) := UC(X, R). We are interested in the set nUC(X, Y ) := C(X, Y )\UC(X.Y ), and specially in nUC(X) := nUC(X, R) = C(X)\UC(X), which is nonempty whenever Xis a non-Atsuji metric space. For a metrizable topological space X, we shall also consider the set BC(X) of all bounded continuous functions X→Ras well as its complement nBC(X) := C(X)\BC(X), which by Hewitt’s theorem is nonempty if and only if Xis non-compact. A topological space Yis said to be an absolute extensor if, for every normal topological space X, every closed subset A⊂Xand every f∈C(A, Y ), there exists F∈C(X, Y ) such that F|A=f. Under this terminology, Tietze’s extension theorem (see, e.g., [40, pp. 219–222]) asserts that Ris an absolute extensor. As another example, any product space IJ–where Jis a nonempty set and Iis an interval of the real line– is an absolute extensor and, in general, any product space Qj∈JYjof absolute extensors is also an absolute extensor: see [36, VII.8.34]. Recall that every metrizable space is normal. A linear topological space is called a Fr´echet space if it is locally convex and completely metrizable. It is well known that the topology of every Fr´echet space Xis defined by a distance d(x, y) = kx−yk, where k·k is an F-norm on X, that is, a functional k · k :X→[0,+∞) satisfying the following properties for all x, y ∈Xand all scalars λ:kx+yk≤kxk+kyk, kλxk≤kxkif kλk ≤ 1, limλ→0kλxk= 0, and kxk= 0 implies x= 0; see [30]. In the following theorem we collect a number of well known results about σ-compactness and the compact-open topology on a space of continuous functions; see e.g. [24, pp. 325–326, 342 and 405–406] or [20]. Recall that a 4 AYALA, BERNAL, CALDER ´ ON, AND VILCHES topological space is called σ-compact if it is a countable union of compact subspaces, and second-countable if it possesses a countable open basis. For background on nets, see, e.g., [32]. Theorem 2.1. Assume that Xis a T2-locally compact topological space and that (Y, ρ)is a metric space. The following holds: (a) If Xis second-countable then it is metrizable and σ-compact. (b) Xis σ-compact if and only if there is a sequence {Un}n≥1of open sets such that X=Sn≥1Un, each Unis compact and Un⊂Un+1 for all n∈N. (c) If Xis σ-compact then there is a metric Don C(X, Y )generating the compact-open topology τK, that is, the metric Dsatisfies the following property: A net (fα)α∈J⊂C(X, Y )tends to f∈C(X, Y ) uniformly on each compact subset of Xif and only if D(fα, f)→0 (α∈J). In fact, under the assumptions of (c) and the notation in (b), a distance in C(X, Y ) generating the compact-open topology is D(f, g) = ∞ X n=1 1 2n·supx∈Unρ(f(x), g(x)) 1 + supx∈Unρ(f(x), g(x)). Moreover, it is easy to prove that if Yis complete then Dis complete. Hence (see, e.g., [42]) (C(X, Y ), τK) becomes a Baire space provided that Yis complete and Xis T2, locally compact and σ-compact. Sometimes the problem of “gluing” two uniformly continuous functions on two respective subsets of a topological space arises naturally. Not always this operation yields a uniformly continuous function, even if the global function is continuous. For instance, if X={1,−1,1/2,−1/2,1/3,−1/3, . . . },A= {1/n :n∈N}and B={−1/n :n∈N}, then X=A∪B, the function f: (X, de)→(R, de) given by f(x) = 0 if x∈A 1 if x∈B is continuous and both restrictions f|A, f|Bare uniformly continuous, but fis not uniformly continuous. In general, we can say that if (X, d) and (Y, ρ) are metric spaces and that Aand Bare subsets of Xsuch that X=A∪Band d(A, B)>0 then, trivially, every function f:X→Y with f|Aand f|Buniformly continuous is uniformly continuous. But this is a rather uncommon situation. More interesting and useful is the following result, which will be used in Section 3 and follows from [26, Corollary 16] as a special case. (Incidently the property of frelative to Ain [26, Corollary 16] is called strong uniform continuity of fon A; this variational notion has been studied in depth by Beer and Levi.) NON-UNIFORMLY CONTINUOUS FUNCTIONS AND UNBOUNDED FUNCTIONS WITHIN C(X)5 Lemma 2.2. Assume that (X, d)and (Y, ρ)are metric spaces and that f∈C(X, Y ). Let A, B ⊂Xbe subsets such that X=A∪B,Ais compact and f|Bis uniformly continuous. Then f∈UC(X, Y ). Some terminology extracted from the modern theory of lineability (see [3, 14] for concepts and results) will be also needed. Assume that αis a cardinal number, Zis a vector space and A⊂Z. Then Ais said to be lineable if there is an infinite dimensional vector space Msuch that M\{0} ⊂ A, and α-lineable if such an Mcan be found so as to satisfy dim (M) = α. If, in addition, Zis a topological vector space, then the subset Ais said to be spaceable (α-dense-lineable, resp.) in Zwhenever there is a closed infinite dimensional (a dense, resp.) vector subspace Mof Zsuch that M\ {0} ⊂ A(such that dim (M) = αand M\ {0} ⊂ A, resp.). Now, assume that Zis a vector space contained in some (linear) algebra. Then the subset Ais called algebrable if there is an infinitely generated algebra M–that is, the cardinality of any system of generators of Mis infinite– so that M\ {0} ⊂ A; and, if αis a cardinal number, then Ais said to be strongly α-algebrable if there exists an α-generated free algebra Mwith M\ {0} ⊂ A. Recall that if Zis contained in a commutative algebra, then a set B⊂Zis a generating set of some free algebra contained in Aif and only if for any N∈N, any nonzero polynomial Pin N variables without constant term and any distinct f1, . . . , fN∈B, we have P(f1, . . . , fN)6= 0 and P(f1, . . . , fN)∈A. The following auxiliary assertion will be useful in order to achieve denselineability from mere lineability. Its proof can be found in [13, Theorem 2.3] (see also [4, Theorem 2.2 and Remark 2.5] and [3, Section 7.3]). Theorem 2.3. Assume that Zis a metrizable separable topological vector space and that γis an infinite cardinal number. Suppose that Aand B are subsets of Zsuch that A+B⊂A,A∩B=∅,Bis a dense vector subspace of Z, and Ais γ-lineable. Then Ais γ-dense-lineable in Z. Note that since the space Zis metrizable and separable, its cardinality satisfies card(Z)≤cand hence dim(Z)≤c. Then c-(dense-)lineability is the optimal degree of (dense-)lineability that one can expect for a subset A⊂Z. In order to construct large algebras of functions satisfying special properties, we shall employ the next lemma, that is inspired by the method used in [21] based on the superpositions of a fixed function belonging to the considered class with the representatives of some well chosen algebra of functions. Lemma 2.4. Let Ωbe a nonempty set, αbe a cardinal number and Fbe a family of functions Ω→R. Assume that there exist an α-generated free algebra Φconsisting of analytic functions R→Rand a function f: Ω →R 6 AYALA, BERNAL, CALDER ´ ON, AND VILCHES satisfying that f(Ω) has an accumulation point in Rand ϕ◦f∈ F for every ϕ∈Φ\ {0}. Then Fis strongly α-algebrable. Proof. Consider the set A:= {ϕ◦f:ϕ∈Φ}, that is an algebra contained in F ∪{0}. Assume that Bis a free generator system for Φ with card (B) = α. Trivially, e B:= {ϕ◦f:ϕ∈ B} is a generator system for A. Moreover, card ( e B) = α= card (B) because ϕ◦f6=ψ◦fas soon as ϕ6=ψ(ϕ, ψ ∈Φ). Indeed, if ϕ◦f=ψ◦fthen ϕ=ψon the set f(Ω), which has an accumulation point in R, and so ϕ=ψon the whole Rby the Identity Principle for analytic functions (see, e.g., [1]). Finally, e Bis a free generator system. To prove this, assume that N∈N, that Pis a polynomial of N real variables and that fϕ1,..., fϕNare distinct functions in e Bsatisfying P(fϕ1,..., fϕN) = 0 on Ω. There are mutually different functions ϕ1, . . . , ϕN∈ B with eϕi=ϕi◦f (i= 1, . . . , N). Therefore Q◦f= 0 on Ω, where Q:= P(ϕ1, . . . , ϕN). It follows that Q= 0 on f(Ω), and again the Identity Principle tells us that Q= 0 on R, which implies P= 0 because Bis algebraically free. Thus, Ais an α-generated free algebra, which concludes the proof.  Remark 2.5. Explicit examples in which the technique of the last lemma (with α=c) is applied are provided in [2,6–8,21]. In [12] a complex version of such technique is used. We need for our goals the family Φ considered in [6]. In the next lemma we list the properties of Φ that will be used in Section 3. Lemma 2.6. There exists an algebra Φof functions R→Rsatisfying the following properties: (a) Φ is freely c-generated. (b) Every ϕ∈Φis analytic on R. (c) For every ϕ∈Φ\ {0}, we have limx→+∞|ϕ(x)|= +∞. (d) Φ is stable under derivations, that is, if ϕ∈Φthen ϕ0∈Φ. Proof. Let H⊂(0,+∞) be a linearly Q-independent set satisfying card (H) = c. For each c∈H, we denote ϕc(x) := ecx. Let Φ be the algebra generated by {ϕc}c∈H. On the one hand, (b) follows from the facts that every ϕcis analytic on Rand the analytic functions on Rform an algebra. On the other hand, (a) is a consequence of (c). And (d) is easily deduced from the fact that ((ϕc1)j1· · · (ϕcN)jN)0= (c1j1+· · · +cNjN)ϕc1j1+···+cNjNfor all reals c1, . . . , cNand all natural numbers j1, . . . , jN. Therefore, we only have to prove (b). With this aim, let us represent each N-tuple (r1, . . . , rN)∈RNby r, and set |r|:= r1+· · · +rNand r·s:= r1s1+· · · +rNsN. Assume that ϕ∈Φ\ {0}. Then there exist an N∈N, mutually different c1, . . . , cN∈H and a nonzero polynomial Pin Nreal variables such that Pdoes not NON-UNIFORMLY CONTINUOUS FUNCTIONS AND UNBOUNDED FUNCTIONS WITHIN C(X)7 possess constant term and ϕ=P(ϕc1, . . . , ϕcN). Specifically, there exists a nonempty finite set J⊂NN 0\ {(0,0,...,0)}and scalars αj∈R\ {0}such that P(x1, . . . , xN) = Pj∈Jαjxj1 1· · · xjN N. Hence ϕ(x) = X j∈J αjec·jxfor all x∈R. Let kbe the unique element in Jsuch that c·k= max{c·j:j∈J}. Note that kis unique due to the linear Q-independence of the cj’s. Then c·j−k·c<0 for all j∈J\ {k}. Of course, j·c>0 for all j∈J. Finally, the conclusion of (b) is obtained from the inequality |ϕ(x)| ≥ ec·kx·|αk| − X j∈J\{k} |αj|e(c·j−c·k)x and the facts that limx→+∞ec·kx= +∞and limx→+∞e(c·j−k·c)x= 0 for all j∈J\ {k}. 3. Topological and algebraic properties of the set of non-uniformly continuous functions In this section, we shall be able to find a rich algebraic structure inside the subset of non-uniformly continuous functions, assuming that the space of departure is non-Atsuji and satisfies appropriate conditions. Concerning topological size, the next theorem shows the denseness of the mentioned subset and the existence of some “reasonable” structure in it. Recall that if Zis a topological space and A⊂Z, then Ais called a Gδ set if Ais a countable intersection of open subsets of Z, an Fσset if it is a countable union of closed subsets of Z, a Gδσ set if it is a countable union of Gδsubsets, and an Fσδ set if it is a countable intersection of Fσsubsets. Recall that τKdenotes compact-open topology. The first three parts of the following theorem are surely known, but since we have not been able to find a reference we provide a proof of them. Theorem 3.1. Assume that (X, d)and (Y, ρ)are metric spaces, and that Xis locally compact and σ-compact. We have: (a) UC(X, Y )is an Fσδ subset of (C(X, Y ), τK). (b) nUC(X, Y )is a Gδσ subset of (C(X, Y ), τK). (c) If Yis either an absolute extensor or a Fr´echet space over Rthen UC(X, Y )is dense in (C(X, Y ), τK). In particular, UC(X)is dense in (C(X), τK). (d) If (X, d)is a non-Atsuji space and Yis a Fr´echet space over R whose metric ρis induced by an F-norm defining the topology of Y, then nUC(X, Y )is a dense subset of (C(X, Y ), τK). In particular, nUC(X)is dense in (C(X), τK). 8 AYALA, BERNAL, CALDER ´ ON, AND VILCHES Proof. (a) Note first that the set UC(X, Y ) can be written as UC(X, Y ) = \ n∈N[ k∈N Fn,k,(1) where Fn,k := {f∈C(X, Y ) : ρ(f(x), f(y)) ≤1 nfor all (x, y)∈X×X with d(x, y)≤1 k}. If (fj)⊂Fn,k is a sequence tending to some fof the (metrizable, by Theorem 2.1) space C(X, Y ), then from the continuity of the map ρ:Y×Y→Rand the fact that τK-convergence implies pointwise convergence it follows that f∈Fn,k. Hence each set Fn,k is closed, which together (1) tells us that UC(X, Y ) is an Fσδ set. (b) This is a consequence of (a) because the complement of an Fσδ subset is aGδσ subset. (c) Recall that the sets V=V(f, L, ε) := {g∈C(X, Y ) : ρ(f(x), g(x)) < ε for all x∈L} (f∈C(X, Y ), ε > 0, Ka compact subset of X) form a basis for the τK- topology of C(X, Y ). Fix such a subset V. In order to prove the denseness of UC(X, Y ), we search for some g∈U∩UC(X, Y ). With this aim, consider the sequence {Un}n≥1of open sets such that X=Sn≥1Un, each Unis compact and Un⊂Un+1 (n∈N) provided by Theorem 2.1(b). Since {Un}n≥1is in particular increasing, we can select an m∈Nwith K⊂Um. Now, Xis normal because it is metrizable. Assume that Yis an absolute extensor. Fix any point y0∈Y. Since the closed sets Um, X \Um+1 are disjoint, the function Um∪(X\Um+1)→Rgiven by fon Umand y0 on X\Um+1 is well defined and continuous. Then there exists a function g∈C(X, Y ) such that g|Um=fand g|X\Um+1 =y0. The same conclusion holds if Yis a real Fr´echet space because any metrizable space is paracompact (see, e.g., [40, Theorem 41.4]), and any continuous function on a closed subset (with the restricted topology) of a Hausdorff paracompact space Zwith values in a real Fr´echet space extends to a continuous function on Z(see [36, X.3.33]). To summarize, we have that gis continuous, Um+1 is compact, g|X\Um+1 is constant (so g|X\Um+1 is uniformly continuous) and X=Um+1 ∪(X\Um+1). Therefore Lemma 2.2 applies and we conclude that g∈UC(X, Y ). But g=fon Kbecause Um⊃Um⊃K. Thus, trivially, g∈Vand g∈V∩UC(X, Y ), as required. (d) By assumption, we have that ρ(x, y) = kx−yk, where k·k is an F-norm on Ydefining its topology. From the hypothesis we can select a continuous function f0:X→Rthat is not uniformly continuous. Fix any nonzero vector y0∈Y, and consider the mapping f:X→Ygiven by f(x) := f0(x)y0. It is evident that f∈C(X, Y ). Then f∈nUC(X, Y ): indeed, there are sequences (un) and (vn) in Xand an α > 0 such that d(un, vn)→0 as n→ ∞ but |f0(un)−f0(vn)| ≥ αfor all n∈N. NON-UNIFORMLY CONTINUOUS FUNCTIONS AND UNBOUNDED FUNCTIONS WITHIN C(X)9 Since k·k is an F-norm and |α/(f0(un)−f0(vn))| ≤ 1, it follows that ρ(f(un), f(vn)) = kf(un)−f(vn)k=k(f0(un)−f0(vn)) y0k≥kα y0k>0 for all n∈N, which proves the claim. Now, from the statement (c) we have that UC(X, Y ) is dense in (C(X, Y ), τK). But C(X, Y ) is a vector space and UC(X, Y ) is a vector subspace of C(X, Y ), which implies that the set of translates f+UC(X, Y ) := {f+g:g∈UC(X, Y )} satisfies f+UC(X, Y )⊂nUC(X, Y ). Since (C(X, Y ), τK) is a topological vector space, we conclude that the translation g∈C(X, Y )7−→ f+g∈C(X, Y ) is a homeomorphism, from which it follows that f+UC(X, Y ) is dense in (C(X, Y ), τK), and so its superset nUC(X, Y ) is also dense.  The next two theorems assert algebrability, dense-lineability and spaceability for the set nUC(X) in a high degree. No extra condition will be needed to guarantee algebrability. Recall that a subset Aof a topological space Xis said to be discrete whenever A0=∅or, that is equivalent, the restricted topology on Ais the discrete one. Theorem 3.2. Let (X, d)be a non-Atsuji metric space. Then the set nUC(X)is strongly c-algebrable. Proof. We are going to built a non-uniformly continuous function f:X→R satisfying that there exist three mutually disjoint sequences (xn),(yn),(zn)⊂ Xsuch that f(xn) = n, f(yn) = n+ 1, f(zn) = 1 nfor all n∈Nand d(xn, yn)→0 as n→ ∞. Assume that this has already been done. Let Φ be the family furnished in Lemma 2.6, and take ϕ∈Φ\ {0}. According to assertions (c) and (d) of the mentioned lemma, we have that limx→+∞|ϕ0(x)|= +∞. Then there exists x0∈Rwith |ϕ0(x)| ≥ 1 for all x≥x0. It follows from the mean value theorem that |(ϕ◦f)(xn)−(ϕ◦f)(yn)|=|ϕ(n)−ϕ(n+ 1)| ≥ 1· |n−(n+ 1)|= 1 provided that n≥x0. Therefore d(xn, yn)→0 but de((ϕ◦f)(xn),(ϕ◦f)(yn)) 6−→ 0 as n→ ∞. This tells us that ϕ◦f∈nUC(X). Observe that, since f(zn) = 1 n(n∈N), the set f(X) has 0 as a limit point. It now suffices to apply Lemma 2.4 with Ω := X,α:= cand F:= nUC(X). Thus, our unique task is to construct a function fas claimed at the beginning of the proof. According to Theorem 1.1, we have that at least one of the following assertions holds: (i) The set X0is not compact. (ii) There exists α > 0 such that, for every δ > 0, there are x, y ∈X with d(x, X0)≥α≤d(y, X0) and 0 < d(x, y)< δ. 16 AYALA, BERNAL, CALDER ´ ON, AND VILCHES 3. Finally, and turning back to a general locally compact σ-compact non- Atsuji metric space (X, d), we also can find large algebraic structures in the family of bounded non-uniformly continuous functions BC(X)∩nUC(X). Namely, this set is spaceable in the (Banach) space BC(X) endowed with the supremum norm k · k∞, and c-dense-lineable in (C(X), τK). In order to see this, notice that if in the proof of Theorem 3.3(b) we had take the functions gk:Fk→R(k∈N) given by gk(x) =    0 if x=xn, n ∈Nk 1 if x=yn, n ∈Nk 0 if x∈Ck instead of the ones provided in the mentioned proof, then we would have obtained extensions fk∈C(X) that are bounded (in fact, with kfkk∞= sup{|gk(x)|:x∈Fk}= 1) thanks to Tietze’s extension theorem for bounded functions (see, e.g., [15, p. 91]). An application of Nikolskii’s theorem for Banach spaces (see, e.g., [19, pp. 36–38]) leads easily to the conclusion that (fk) is a basic sequence. Therefore its closed linear span Min (BC(X),k · k∞) happens to be an infinite dimensional closed vector subspace, and arguing as in the ending part of the proof of Theorem 3.3(b) one can conclude that M\{0} ⊂ nUC(X). This shows the claimed spaceability. Now, a well-known consequence of the Baire category theorem tells us that dim (M) = c. Hence A:= BC(X)∩nUC(X) is γ-lineable, where γ:= c. By Theorem 3.1(c), the set UC(X) is dense in (C(X), τK), but an analysis of its proof reveals –again via Tietze’s extension theorem for bounded functions– that the approximating function g∈V∩UC(X, Y ) (with Y=R) can be taken bounded. In other words, the set B:= BC(X)∩UC(X) is dense in C(X). Moreover, since BC(X) and UC(X) are vector spaces, we obtain that Bis a dense vector subspace of Z:= C(X) and A+B⊂A. Of course, A∩B=∅. The claimed c-dense-lineability of Afollows after an application of Lemma 2.3. We want to pose here the following question: Is BC(X)∩nUC(X) (strongly)algebrable? Acknowledgements. 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Departamento de Geometr´ ıa y Topolog´ ıa Facultad de Matem´ aticas Universidad de Sevilla Avenida Reina Mercedes, Sevilla, 41012, Spain E-mail address:[email protected] Departamento de An´ alisis Matem´ atico Facultad de Matem´ aticas Instituto de Matem´ aticas Antonio de Castro Brzezicki (IMUS) Universidad de Sevilla Avenida Reina Mercedes, Sevilla, 41012, Spain E-mail address:[email protected] Departamento de An´ alisis Matem´ atico Facultad de Matem´ aticas Instituto de Matem´ aticas Antonio de Castro Brzezicki (IMUS) Universidad de Sevilla Avenida Reina Mercedes, Sevilla, 41012, Spain E-mail address:[email protected] NON-UNIFORMLY CONTINUOUS FUNCTIONS AND UNBOUNDED FUNCTIONS WITHIN C(X)19 Departamento de Geometr´ ıa y Topolog´ ıa Facultad de Matem´ aticas Universidad de Sevilla Avenida Reina Mercedes, Sevilla, 41012, Spain E-mail address:[email protected]