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For each s ∈ (0; 1], we give an example of a nowhere dense perfect set E contained in the unit interval with dimH(E) = s, which is not an attractor for any iterated function system composed of weak contractions. This answers a problem posed by Zoltán Buczolich at the Summer Symposium in Real Analysis XXXIX (June 8-13, 2015, St. Olaf College, Northfield, MN). [ABSTRACT FROM AUTHOR]

For any map f from ℝ to ℝ, if an interval J contains all periodic points of period 1 and 2, then f(f(J)) contains all periodic points (and therefore contains the centre of f). [ABSTRACT FROM AUTHOR]

There is a function F : [c]<ω → {0,1} such that if A ⊆ [c]<ω is uncountable, then {F(a ∪ b) : a, b ∈ A, a ≠ b} = {0, 1}. A corollary is that there is a function f : ℝ → {0, 1} such that if A ⊆ ℝ is uncountable, 2 ≤ k < ω, then both 0 and 1 occur as the value of f at the sum of k distinct elements of A. This was originally proved by Hindman, Leader, and Strauss under CH, and they asked if it holds in general. [ABSTRACT FROM AUTHOR]

Answering one of the real function problems suggested by A. Maliszewski, the existence of a bounded Darboux function of the Sierpiński first class that cannot be expressed as a difference of two bounded lower semicontinuous functions is proved. As the reply to the other Maliszewski question, we show there exists an almost everywhere continuous Darboux function of the Sierpiński first class that is not a difference of two almost everywhere continuous lower semicontinuous functions. [ABSTRACT FROM AUTHOR]

We provide a characterization of lower semicontinuity for multifunctions with values in a metric space (Y, d) which, in the special case of single-valued functions, says that a function is continuous if and only if for each ε > 0, the ε-tube about its graph is an open set. Applications are given, one of which provides a novel understanding of the Open Mapping Theorem from functional analysis. We also give a related but more complicated characterization of upper semicontinuity for multifunctions with closed values in a metrizable space [ABSTRACT FROM AUTHOR]

The purpose of this article is to give a solution to a problem raised by A. Maliszewski in [5] by showing that any lower semicontinuous function can be represented as a sum of two lower semicontinuous strong Šwiątkowski functions. [ABSTRACT FROM AUTHOR]

Some additive properties of the following three families of "pathological" functions are briefly discussed: continuous nowhere differentiable functions, Sierpiński-Zygmund functions, and absolutely nonmeasurable functions. INSET: '. [ABSTRACT FROM AUTHOR]

The concept of uniform differentiability is introduced to characterize sequences of McShane and Henstock equi-integrable functions. [ABSTRACT FROM AUTHOR]

We prove that for any lower semicontinuous function f : [0, 1] → [0, 1] with purely unrectifiable graph and for any ε > 0 there is an open set U ⊃ graph f with every vertical section set of one-dimensional Lebesgue measure at most ε. [ABSTRACT FROM AUTHOR]

By using the boundedness of the maximal and sharp operators on Morrey spaces, we have proved that the commutators [Mp, b] and [M#, b] are bounded on Morrey spaces Lq, λ if and only if b is in BMO and the negative part of b is in L∝ [ABSTRACT FROM AUTHOR]

We characterize products of Baire one double star functions. [ABSTRACT FROM AUTHOR]

In this paper we give the definition of [λ ᵘ]-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of [λ, ᵘ]-continuous functions. The aim of the paper is to find the maximal additive class and the maximal multiplicative class for the family of [λ, ᵘ]-continuous functions. [ABSTRACT FROM AUTHOR]

Existence of measure preserving maps has been discussed in books on real analysis where the Axiom of Choice is instrumental. In this note we introduce a method to construct such maps. For our construction we use the distribution function and elementary differential equations. [ABSTRACT FROM AUTHOR]

The present paper is a survey on Ck,1 functions. Both theoretical and numerical results related to this class of nonsmooth functions are presented. We also give few new results and pose some open problems for further investigation. [ABSTRACT FROM AUTHOR]

In this paper we give some equivalence relations between behavior of Fourier coefficients of a special kind and smoothness of functions. A necessary and sufficient condition for existence of Schwartz derivative is also obtained. [ABSTRACT FROM AUTHOR]

The aim of the paper is to give sufficient and necessary conditions when the attractor of an iterated multifunction system, composed by multifunctions with compact and connected values, is a connected set. [ABSTRACT FROM AUTHOR]

We show that functions of two variables whose vertical sections are preponderantly continuous are strongly supmeasurable. [ABSTRACT FROM AUTHOR]

The system of functional differential equations (1) has a continuously differentiable solution for every value of the parameter a. The boundary values and a are related with d(2 - a) = c(2 + a). When a ∈ S where S= {22n+1 :n=1,2,3, …}, the system (1) has infinitely many solutions with boundary values c= 0 and d = O. For all other values of a, the system (1) has a unique solution. F' (x) = aF(2x) if 0 ≤ x ≤ ½ F' (x) = aF(2-2x) if ½ x ≤ 1 F(0) = c, F(1) = d. [ABSTRACT FROM AUTHOR]

In this paper we begin the search for appropriate definitions of first-return continuity and first-return approachability for functions of several real variables. We do this by first reviewing the one dimensional theory and then considering several candidate definitions in higher dimensions. [ABSTRACT FROM AUTHOR]

A new characterization of subcontinuity of functions is given. Relations among subcontinuity, local boundedness and local compactness are studied. The set of points of subcontinuity of functions is investigated. [ABSTRACT FROM AUTHOR]

We present a generalization of the following result of Y. Benyamini. There is a continuous function f : ℝ → ℝ such that for each (xn)n∈ℤ ∈ [0, 1]ℤ, there is t ∈ ℝ such that xn = f(t + n) for all n ∈ ℤ. [ABSTRACT FROM AUTHOR]

We characterize differentiability of a map f : ℝ → ℝ in terms of continuity of a canonically associated map f̂. To characterize pointwise differentiability of f, both the domain and range of f̂ can be made topological. However, the global differentiability of f is characterized by the continuity of f̂ whose domain is topological but whose range is a convergence space. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two inequalities for generalized rational functions of one variable in Lp spaces when a partition of the domain with a suitable number of measurable subsets is considered. [ABSTRACT FROM AUTHOR]

This paper introduces a notion of equivalence that links closed relations and quasicontinuous functions; we examine classes of quasicontinuous functions that have the same set of continuity points. In doing so, we show that every minimal closed relation is the closure of a quasicontinuous function and vice-versa. We also show that this notion is of use in dynamical systems. Every quasicontinuous function is equivalent to one that is measurable, and under certain circumstances--in fact, under just those circumstances that appear most often in the dynamics literature--it is equivalent to a quasicontinuous function that has an invariant measure. [ABSTRACT FROM AUTHOR]

A function f : ℝ → ℝ is said to be vertically rigid provided its graph G(f) = {〈x, f(x)〉: x ∈ R} is isometric with the graph of the function kf for every non-zero k ∈ ℝ. We show that the group homomorphisms f from 〈ℝ,+〉 into 〈ℝ+, ·〉 is vertically rigid if and only if it is an epimorphism. Some other examples of vertically rigid functions will also be given. A problem of characterizing all vertically rigid functions remains open. [ABSTRACT FROM AUTHOR]

Presents the response of researchers on differential equations in France and Italy to Ivar Fredholm's theory of functional equations involving integrals. Contribution of the theory to the development of functional analysis; Background on the development of integral equations in the nineteenth-century; Role of Henri Poncaré in developing the field of partial differential equations.

Deals with the search for extreme values of a differentiable function. Formula showing the n-th partial derivative of Lebesgue's singular function; Location of the maximum and minimum points of the derivative; Approach used in obtaining the binary expansion of any minimum point.

The purpose of this article is to show that if - ∞ < α < β < ∞, then the set PDαβ of piecewise differentiable functions in C([α, β],ℝ) is π1¹-complete. [ABSTRACT FROM AUTHOR]

Given two topological spaces X and Y and a family 풪* of subsets of X, a function f : X → Y is called 풪*-continuous if f-1 (V ) ∈ 풪* for every open set V ⊆ Y . An 풪*-step function is meant to be a function 훗 : X → Y that is piecewise constant on a partition of X into sets from 풪*. Using some technical assumptions on X, Y , and 풪* we give representations of 풪*-continuous functions as uniform limits of 풪*-step functions. We deal in particular with α-continuous, nearly continuous, almost quasi-continuous, and somewhat continuous functions. The paper is motivated by a corresponding characterization of quasi-continuous functions. [ABSTRACT FROM AUTHOR]

In the paper we present an exhaustive discussion of the relations between Darboux-like functions within the class of additive Sierpiński-Zygmund (SZ) functions. In particular, we give an example of an additive Sierpiński-Zygmund (SZ) injection f : ℝ → ℝ such that f-1 is not an SZ function. Under the assumption that R cannot be covered by less than c-many meager sets we give examples of an additive SZ bijection f : ℝ → ℝR such that f-1 is not SZ and of an additive injection f : ℝ → ℝ such that both f and f-1 are SZ. [ABSTRACT FROM AUTHOR]

Using almost-invariant sets, we show that a family of Marczewski-Burstin algebras over groups are not closed under algebraic sums. We also give an application of almost-invariant sets to the difference property in the sense of de Bruijn. In particular, we show that if G is a perfect Abelian Polish group then there exists a Marczewski null set A ⊆ G such that A + A is not Marczewski measurable, and we show that the family of Marczewski measurable real valued functions defined on G does not have the difference property. [ABSTRACT FROM AUTHOR]

In this note we give, for any open subset in ℝn, a function describing the boundary of this set with exact regularity and being, globally, as regular as possible. [ABSTRACT FROM AUTHOR]

We prove the existence of three subspaces of C[0, 1], each homeomorphic to C[0, 1]; the first consists only of infinitely many times differentiable functions, the second consists only of singular functions of bounded variation, and the third consists only of nowhere differentiable functions. [ABSTRACT FROM AUTHOR]

A density continuous function is defined as a continuous function from a Tychonoff space X into the real numbers with the density topology. The collection of density continuous functions on X is denoted by C(X, ℝd). It is shown that C(X,ℝd) is a ring precisely when each density continuous function is locally constant, and in this case X is defined to be a density P-space. Examples of density P-spaces are given. [ABSTRACT FROM AUTHOR]

Making use of ideas of Marczewski and Sierpiński we propose a general approach to studies on connections between measurability, continuity and relative continuity of functions. Theorem 2.1 shows that a wellknown characterization of (s)-measurable Marczewski functions can be extended to the case of functions measurable with respect to a wide class of algebras involved with a topology. Theorem 2.2 generalizes the Denjoy-Stepanoff theorem and shows that the Denjoy-Stepanoff property stating the continuity of A-measurable functions at all points of a co-negligible set is quite common while an algebra A and an ideal J are the results of operations S and S0 on τ∖ I for a given topology τ . Also from the obtained results we conclude new theorems concerning the algebras associated with product ideals (Theorems 3.12 and 3.13). [ABSTRACT FROM AUTHOR]

Lévy's dragon curve [P. LÉVY, Les courbes planes ou gauches et les surfaces composées de parties semblables au tout, J. Ecole Polytechn., 227-247, 249-291 (1938)] is a well-known self-similar planar curve with non-empty interior. We derive an arithmetic expression for the coordinate functions of Lévy's dragon curve, and show that the 3/2 -dimensional Hausdorff measure of the graph of each coordinate function is strictly positive and finite. This complements known dimensional results concerning the coordinate functions of space-filling curves of Peano and Hilbert. The proof is based on deriving suitable uniform upper bounds for the sizes of the graphs' level sets. [ABSTRACT FROM AUTHOR]

It is shown that if f is Henstock-Kurzweil integrable on a compact interval E in ℝm, then f is upper semicontinuously integrable on E if and only if there exists an increasing sequence {Xn} of closed sets whose union is E, and f¦xn is bounded for each positive integer n. [ABSTRACT FROM AUTHOR]

We construct a C¹-function f : [0, 1] → ℝ such that for almost all x ∈ (0, 1), there is r > 0 for which f(y) > f(x) + f′(x)(y - x) when y ∈ (x, x + r) and f(y) < f(x) + f′(x)(y - x) when y ∈ (x - r, x). The existence of such functions is related to a problem concerning conical density properties of Hausdorff measures on ℝn. We also discuss the tangential behavior of typical C¹-functions, using an improvement of Jarník's theorem on essential derived numbers. [ABSTRACT FROM AUTHOR]

We study quasi-monotonic functions of the Zygmund-Bary-Stechkin class Φ with the main emphasis on properties of the index numbers of functions in this class. A special attention is paid to functions whose lower and upper index numbers do not coincide with each other (non-equilibrated functions). It is proved that the bounds for functions in Φ known in terms of these indices, are exact in a certain sense. We also single out some special family of none-equilibrated functions in Φ which oscillate in a certain way between two power functions. Given two numbers 0 < α ≤ β < 1, we explicitly construct examples of functions in Φ for which α and β serve as the index numbers. The investigation of properties of non-equilibrated functions in Φ was evoked by applications of these properties in problems of the normal solvability of some singular integral operators in the spaces with prescribed modulus of continuity. [ABSTRACT FROM AUTHOR]

A function h : ℝn → ℝk is called a Hamel function if it is a Hamel basis for ℝn+k. We prove that there exists a Hamel function which is finitely continuous (its graph can be covered by finitely many partial continuous functions). This answers the question posted in [3]. [ABSTRACT FROM AUTHOR]

This paper deals with rings of real Darboux functions defined on some topological spaces. Results are given concerning the existence of essential, as well as prime Darboux rings. We prove that, under some assumptions connected with the domain X of the functions, the equalities: D(X) = Slf (X), S(X) = dim(ℜ) hold, where D(X) is a D-number of X, S(X) (Slf (X)) denotes the Souslin (lf-Souslin) number of X and dim(ℜ) is a Goldie dimension of an arbitrary prime Darboux ring ℜ. [ABSTRACT FROM AUTHOR]

A function f : ℝ → ℝ has the property 훭1 (훭2) if the restricted function f↾D(f) (f↾Dap(f)) is monotone. (D(f) [Dap(f)] denotes the set of all discontinuity points [the set of all approximate discontinuity points] of f.) In this article I investigate the uniform, pointwise and transfinite limits of sequences of functions with the property 훭i, i = 1, 2. [ABSTRACT FROM AUTHOR]

In this article we describe the products of two unilaterally continuous regulated functions. [ABSTRACT FROM AUTHOR]

Discusses periodic and recurrent points of continuous functions. Theorems; Difficulties associated with the use of the Baire Category argument.

In the paper of Lee et al. an equivalent condition for a function ƒ to be of the first Baire class has been established. This condition is of an ε -- δ type, similarly as in Cauchy's definition of continuity of a function. In the first part of this paper we examine a problem whether it is possible to obtain other classes of functions by further modifications of the above condition. It turns out that, in some sense, the answer is negative. In the second part we consider a topological version of the condition of Lee et al. [ABSTRACT FROM AUTHOR]

We give a new and simple proof that every polyharmonic function on Rn which is bounded is constant. [ABSTRACT FROM AUTHOR]

A function ƒ : Rm → R is called s1-strongly quasi-continuous at a point X ∈ Rm if for each real ε > 0 and for each set A ∋ x belonging to the density topology, there is a nonempty open set V such that Ø ≠ A ∩ V ⊂ ƒ-1((ƒ(x) - ε, ƒ(x)) ∩ C(ƒ), where C(ƒ) denotes the set of continuity points of ƒ. It is proved that every λ-almost everywhere continuous function ƒ : Rm → R is the quasi-uniform limit of a sequence of s1-strongly quasi-continuous functions and that each measurable function ƒ : Rm → R is the quasi-uniform limit of a sequence of approximately quasi-continuous functions ƒ : Rm → R. [ABSTRACT FROM AUTHOR]

Assuming that the union of fewer than c-many meager sets does not cover the real line, we construct an example of an additive almost continuous Sierpiński-Zygmund function which has a perfect road at each point but which does not have the Cantor intermediate value property. [ABSTRACT FROM AUTHOR]

In 1958 T. M. Flett proved a theorem which is a variant of the Lagrange mean value theorem; namely, let ƒ : [a, b] → R be a differentiable function in [a, b] and ƒ' (a) = ƒ' (b). Then there exists a number η ∈ (a, b) such that ƒ(η) - ƒ(a) = (η - a) - ƒ' (η). Manav Das, Thomas Riedel and Prasanna K. Sahoo have given generalizations of Flett's theorem for approximately differentiable functions. Here we provide generalizations of these theorems for some local S-systems. [ABSTRACT FROM AUTHOR]