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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]

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]

In this paper, we study the stochastic integral equation with its stochastic integral defined using the Henstock approach, or commonly known as the generalized Riemann approach, instead of the classical Itô integral, which we shall call it the Itô-Henstock integral equation. Our aim is to prove the existence of solution of the Itô-Henstock integral equation using the well known method used in the existence theorem of the ordinary differential equation, namely the Picard's iteration method. [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]

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]

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]

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]

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]

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]

The paper is devoted to establishing a necessary condition for the existence of the Riemann-Stieltjes integral of continuous functions and simultaneously to finding a limit form of its relationship with the classical Riemann integral depending on a parameter. The mathematical technique for this study is based on changing the weighted first difference of a given function into the simple difference of a function determined by a first-order difference equation. [ABSTRACT FROM AUTHOR]

The article presents a summary of a paper on a new approach to impulsive retarded functional differential equation (RFDE) presented by speaker Marcia Federson at the Thirty-First Summer Symposium in Real Analysis held at the University of Oxford, England from August 13 to 15, 2007. It cites that it is recognized that impulsive RFDEs can be considered as Banach-space values generalized ordinary differential equations (generalized ODEs). The variational stability and variational asymptotic stability of the zero solution of generalized ODEs are discussed.