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In the paper we will focus on weakly Światkowski functions. We say that f satisfies the weak Świa̧tkowski condition (or f is a weakly Świa̧tkowski function) if for all x 1 , x 2 with f (x 1) < f (x 2) there is a point x between x 1 and x 2 such that f (x 1) < f (x) < f (x 2) . This definition is a modification of the Świa̧tkowski condition, in which point x mentioned above has to be a point of continuity of f. In the paper we will examine some properties of weakly Świa̧tkowski functions and consider lineability and algebrability of some families of functions related to them. [ABSTRACT FROM AUTHOR]

In this paper, using the tools from the lineability theory, we distinguish certain subsets of p-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional algebraic structure: (i) continuously differentiable but not strictly differentiable functions, (ii) strictly differentiable functions of order r but not strictly differentiable of order r + 1 , (iii) strictly differentiable functions with zero derivative that are not Lipschitzian of any order α > 1 , (iv) differentiable functions with unbounded derivative, and (v) continuous functions that are differentiable on a full set with respect to the Haar measure but not differentiable on its complement having cardinality the continuum. [ABSTRACT FROM AUTHOR]

A subset A of a vector space X is called \alpha-lineable whenever A contains, except for the null vector, a subspace of dimension \alpha. If X has a topology, then A is \alpha-spaceable if such subspace can be chosen to be closed. The vast existing literature on these topics has shown that positive results for lineability and spaceability are quite common. Recently, the stricter notions of (\alpha,\beta)-lineability/spaceability were introduced as an attempt to shed light on more subtle issues. In this paper, among other results, we prove some general criteria for the notion of (\alpha,\beta)-lineability/spaceability and, as applications, we extend recent results of different authors. [ABSTRACT FROM AUTHOR]

In this paper, we will focus on different types of Świątkowski conditions: Świątkowski, strong Świątkowski and weak Świątkowski conditions. We present the main properties of the families of functions fulfilling such conditions. [ABSTRACT FROM AUTHOR]

In the paper we will focus on lineability of some subsets of R 0 , 1 which are called linearly sensitive. A function f is called linearly sensitive with respect to the property (or condition) (P) if f has the property (P) and for any a ≠ 0 the function f + a · id does not have the property (P). We discuss some general method of proving c -lineability and use this method to examine lineability of the family of all continuous functions linearly sensitive to the Luzin (N)-property, the family of functions linearly sensitive to the Świątkowski condition and the family of functions linearly sensitive to the strong Świątkowski condition. [ABSTRACT FROM AUTHOR]

Let X be a unitial Banach algebra, G be the set of functions f : [0, 1] → X that are non strongly McShane integrable. The aim of this paper is to show that there is an infinite dimensional closed vector space in G ∪ {0}. [ABSTRACT FROM AUTHOR]

In this paper, we investigate to what extent the conclusion of the Lebesgue dominated convergence theorem holds if the assumption of dominance is dropped. Specifically, we study both topological and algebraic genericity of the family of all null sequences of functions that, being continuous on a locally compact space and integrable with respect to a given Borel measure in it, are not controlled by an integrable function. [ABSTRACT FROM AUTHOR]

In this paper, a criterion for a sequence of composition operators defined on the space of holomorphic functions in a complex domain to be frequently hypercyclic is provided. Such a criterion improves some already known special cases, and, in addition, it is also valid to provide dense vector subspaces as well as large closed ones consisting entirely, except for zero, of functions that are frequently hypercyclic. [ABSTRACT FROM AUTHOR]

It is well known that there are Q -linear discontinuous functions from R to R. Moreover, such functions have dense graph in the real plane. In this paper, we study the existence of linear spaces contained in the family of linear discontinuous functions in the generalized context of topological vector spaces and fields endowed with an absolute value. Furthermore, in the case of fields endowed with an absolute value, we study the algebraic genericity of the families of (1) linear discontinuous functions that are not everywhere surjective and (2) linear discontinuous functions that are strongly everywhere surjective. This extends a number of previous results of lineability theory. Generalizations of classical results of Q -linear discontinuous functions are also analyzed. [ABSTRACT FROM AUTHOR]

For each vector x\in \ell ^{\infty }, we can define the non-empty compact set L_x of accumulation points of x. Given an infinite subset A of \mathbb {N}\backslash \{1\}, we can therefore investigate under which conditions on A, the set L(A)≔\{x\in \ell ^\infty : |L_x|\in A\} is lineable or even densely lineable. In particular, we show that if L(A) is lineable then there exists k\ge 1 such that A\cap (A-k) is infinite and that if L(A) is densely lineable then A\cap (A-1) is infinite. We end up by answering an open question on the existence of a closed non-separable subspace in which each non-zero vector has countably many accumulation points. [ABSTRACT FROM AUTHOR]

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors. [ABSTRACT FROM AUTHOR]

The class D of generalized continuous functions on R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property. The algebra A (D) of classes of functions generated by these families contains 17 atoms. In this work we will calculate the values of the additivity coefficient A (F) for all atoms F in the algebra A (D) . We also determine the values A (F) for a lot of other families F ∈ A (D) . Open questions and new directions of research shall also be provided. [ABSTRACT FROM AUTHOR]

In this paper, we continue the ongoing research on lineability and algebrability within the context of p-adic analysis. This work serves a twofold purpose. Firstly, we improve and, at times, generalize some already known results within this fields and, secondly, we present some new results which have no counterpart in the classical Archimedean framework. [ABSTRACT FROM AUTHOR]

In this paper, the notion of [ S ] -lineability (originally coined by Vladimir I. Gurariy) is introduced and developed in a general abstract setting. This new notion is, then, applied to specific situations, as for instance, classes of differentiable nowhere monotone functions as well as families of vectors having dense orbit with respect to an operator. Large convex structures are also shown to exist inside the family of topologically mixing continuous selfmaps of a real compact interval. [ABSTRACT FROM AUTHOR]

The original version of the celebrated Bloch–Nevanlinna problem asked whether there exists or not a holomorphic function in the unit disc of bounded characteristic whose derivative is not of bounded characteristic. This problem has been solved in the affirmative by a number of mathematicians. Starting from a result on topological genericity of this class of functions due to Hahn, our work intends—under an algebraic as well as a topological point of view—to contribute to this topic. Specifically, the family of solutions of the Bloch–Nevanlinna problem is proved to be residual in appropriate topological spaces, and it contains, except for the zero function, dense maximal dimensional vector subspaces, large linear algebras and infinite-dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract The class Ext of all extendable functions from R to R is the smallest among all Darboux-like classes of functions, which constitute different natural generalizations of the class of usual continuous functions. The goal of this paper is to construct, within Ext, an algebra A which has maximal possible cardinality, that is, 2 c. This, in particular, would confirm a conjecture of T. Natkaniec from 2013. Moreover, the constructed algebra A consists only of functions that are both Baire and Lebesgue measurable. [ABSTRACT FROM AUTHOR]

In this paper we establish that the set of Lipschitz functions f : U → ℝ (U a nonempty open subset of ℓd¹) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of ℓ∞(N)). This result follows along a similar line to that of a previous result of Borwein and Wang (see [Proc. Amer. Math. Soc., 128 (2000), pp. 3221-3229; Bull. Aust. Math. Soc., 72 (2005), pp. 491-496]). However, while the latter was based on Baire's category theorem, our current approach is constructive and is not linked to uniform convergence. In particular, we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in [ 0 , + ∞) and of the family of sequences of these functions converging to zero uniformly on compacta and in L 1 -norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero. [ABSTRACT FROM AUTHOR]

The Borel map j ∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. In the literature, it is well known that the restriction of j ∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In this paper, we are interested in studying how large the image of j ∞ is and we investigate the size and the structure of this image by using different approaches (Baire residuality, prevalence and lineability). We give an answer to this question in the very general setting of quasianalytic ultradifferentiable classes defined by weight matrices, which contains as particular cases the classes defined by a single weight sequence or by a weight function. [ABSTRACT FROM AUTHOR]

In this short note, we will study the existence of a vector space of continuous functions f : Z p → Q p , where Z p and Q p are, respectively, the ring of p-adic integers and the field of p-adic numbers, such that each nonzero function does not satisfy the Luzin (N) property and the dimension of the vector space is the continuum. [ABSTRACT FROM AUTHOR]

K. Weierstrass (1872) was probably the first to present the existence of continuous nowhere differentiable functions (although B. Bolzano, in 1822, was the first to come up with such a construction). Almost a century later, V. Gurariĭ [Dokl. Akad. SSSR 167 (1966), pp. 971–973] observed that the family of continuous functions on [0,1] that are differentiable at no point contains, except for the null function, an infinite dimensional vector space. Moreover, and among other recent contributions in this direction, S. Hencl [Proc. Amer. Math. Soc. 128 (2000), p. 3505–3511] generalized the previously mentioned result by proving the existence of isometrical embeddings of separable Banach spaces into the set of nowhere approximatively differentiable and nowhere Hölder functions. Here, we continue this ongoing research with the study of continuous nowhere Hölder functions, no longer defined in subsets of \mathbb {R}, but in subsets of the p-adic field \mathbb {Q}_p. [ABSTRACT FROM AUTHOR]

In this paper we study the divergence behavior of linear approximation processes in general Banach spaces. We are interested in the structure of the set of vectors creating divergence. The Banach–Steinhaus theory gives some information about this set, however, it cannot be used to answer the question whether this set contains subspaces with linear structure. We give necessary and sufficient conditions for the lineability and the spaceability of the set of vectors creating divergence. [ABSTRACT FROM AUTHOR]