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.

This paper addresses conditions for the Abel method of limitability to imply convergence and subsequential convergence.

In this paper we define the ap-Denjoy integral and show that the ap-Denjoy integral is equivalent to the ap-Henstock integral and the integrals are equal.

The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis...

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15],...

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.

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given increasing homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under $\varphi$ and which contains in its interior no fixed point except for $1$. They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if ${ʃ}_{\Omega }xyd\mu \le {\varphi }^{-1}\left({ʃ}_{\Omega }\varphi \circ xd\mu \right){\psi }^{-1}\left({ʃ}_{\Omega }\psi \circ xd\mu \right)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist...

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. A characterization of the class $𝒮(J,\varphi )$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when $\varphi$ is increasing. In the present paper we solve the converse problem, for which continuous maps $f(0,\infty )\to J$, where $J$ is an interval, there is an increasing homeomorphism $\varphi$ of $J$ such that $f\in 𝒮(J,\varphi )$. We...

We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected $G_{\delta }$ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $\left[a,b\right]$ into a Banach space $X.$ It is shown that a Denjoy-Bochner integrable function on $\left[a,b\right]$ is Denjoy-Riemann integrable on $\left[a,b\right]$, that a Denjoy-Riemann integrable function on $\left[a,b\right]$ is Denjoy-McShane integrable on $\left[a,b\right]$ and that a Denjoy-McShane integrable function on $\left[a,b\right]$ is Denjoy-Pettis integrable on $\left[a,b\right].$ In addition, it is shown that for spaces that do not contain a copy of $c_0$, a measurable Denjoy-McShane integrable...