This paper is devoted to give a new method of generating T-equivalence using shape function and finding the exact calculation formulas of T-equivalence induced by shape function on the real line. Some illustrative examples are given.

For a given function in some classes related to real derivatives, we examine the structure of the set of points which are not Lebesgue points. In particular, we prove that for a summable approximately continuous function, the non-Lebesgue set is a nowhere dense nullset of at most Borel class 4.

In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral ${\int }_a^{b}f\mathrm{d}g$ exists if $f\in {\mathrm{B}V}_{\phi }[a,b]$, $g\in {\mathrm{B}V}_{\psi }[a,b]$ and ${\sum }_{n=1}^{\infty }{\phi }^{-1}(1/n){\psi }^{-1}(1/n)<\infty$. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.

According to a result of Kočinac and Scheepers, the Hurewicz covering property is equivalent to a somewhat simpler selection property: For each sequence of large open covers of the space one can choose finitely many elements from each cover to obtain a groupable cover of the space. We simplify the characterization further by omitting the need to consider sequences of covers: A set of reals X has the Hurewicz property if, and only if, each large open cover of X contains a groupable subcover. This...

We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ${\left(V_x\right)}_{x\in X}$ of X such that d(f(x),f(y)) < ε whenever $(x,y)\in V_y\times V_x$. We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.

In the paper the existing results concerning a special kind of trajectories and the theory of first return continuous functions connected with them are used to examine some algebraic properties of classes of functions. To that end we define a new class of functions (denoted $Con{n}^{*}$) contained between the families (widely described in literature) of Darboux Baire 1 functions (${\mathrm{DB}}_1$) and connectivity functions ($Conn$). The solutions to our problems are based, among other, on the suitable construction of the ring,...

It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map $T_a$ with slope a is dense in the interval of transitivity of $T_a$. We prove that the complement of this set of parameters of full measure is σ-porous.

Among the many characterizations of the class of Baire one, Darboux real-valued functions of one real variable, the 1907 characterization of Young and the 1997 characterization of Agronsky, Ceder, and Pearson are particularly intriguing in that they yield interesting classes of functions when interpreted in the two-variable setting. We examine the relationship between these two subclasses of the real-valued Baire one defined on the unit square.