In this note we prove a controlled convergence theorem for the AP integral by using a new version of generalized absolutely continuous functions.

We prove a Harnack extension and a form of the generalized dominated convergence theorem for the AP integral.

In this paper we prove that the extreme partial derivatives of a function having the Baire property, have the Baire property too.

In the paper we study transformations from ${ℝ}^2$ into $ℝ$ and from ${ℝ}^2$ into ${ℝ}^2$ continuous with respect to different density topologies on the domain and range. In the former case the range will always be equipped with the one-dimensional density topology and the domain with either ordinary density or strong density topology. In the later case the domain and the range will be equipped simultaneously with the ordinary density or strong density topology. We will investigate the relations between these classes and with the class of ordinary continuous transformations. We will also examine relation between the (strong) density continuity of ${ℝ}^2\to ℝ$ and of $f(\cdot ,y),f(x,\cdot )$. Similar question will be considered for transformations $F=(f,g):{ℝ}^2\to {ℝ}^2$ and their coordinate functions $f$ and $g$.

A necessary and sufficient condition is given that the limit $f$ of a sequence $\left\{f_n\right\}$ of generalized Riemann integrable functions be integrable and that ${\int }_{I}f=\lim {\int }_I{f}_n$.

The structure of minimal attraction centers of trajectories of continuous maps of the interval is investigated. It is proved that a closed subset $S$ of the interval is the minimal attraction center of a trajectory of a continuous map of the interval into itself if and only if $S$ is either a nowhere dense set or a union of finitely many mutually disjoint nondegenerate closed intervals.

Continuous functions of bounded $\mathrm{\lambda }$-variation that are differentiable at least at one pointform a dense set of first Baire category in $C\mathrm{\Lambda }BV$, the Banach space of continuous functions of bounded $\mathrm{\lambda }$-variation. An example of a nowhere differentiable continuous function of bounded $\mathrm{\lambda }$-variation is given. Furthermore, $C\mathrm{\Lambda }BV$, as a subset of $C[0,1]$ with the usual sup-norm, is a dense subset of first Baire category.

According to Wilczyński’s scheme, we consider density points for product $\sigma$-ideals $𝕂\times 𝕃$ and $𝕃\times 𝕂$. We get the properties analogous to Lebesgue density theorem and create the respective topologies.

A function $f:X\to ℝ$ is simply continuous if for each open set $V$ in $ℝ$, the set $f^{-1}(V)$ is the union of open and a nowhere dense set in $X$. The maximal additive and maximal multiplicative family for the class of all simply continuous functions is investigated.

We show that typical Lipschitz functions on $[0,1]$ (more precisely, typical functions in the space of functions on $[0,1]$ with Lipschitz constant at most one endowed with the topology of uniform convergence) have much better differentiability behaviour then general Lipschitz functions.

In this paper we investigate the problem of extending Darboux functions with finite varation in such a way that the sets of points of quasi-continuity are maintained.

In this sequel it is proved that each upper semicontinuous function $f$ is the sum of two Darboux upper semicontinuous quasi-continuous functions. It is also shown that we can moreover require that the summands preserve all points of continuity of $f$, one of them is non-positive and Lebesgue, and that they are bounded provided that $f$ is bounded.

Let $f:[a,b]\to ℝ$ be a nonnegative and nondecreasing function and let $x_i:[a,b]\to ℝ(i=1,\dots ,n)$ be nonnegative and nondecreasing functions with continuous first derivative. If , then

$${\left({\displaystyle {\int }_b^a{\left({\left({\displaystyle \sum _{i=1}^n{x}_i\left(t\right)}\right)}^p\right)}^{\prime }f\left(t\right)dt}\right)}^{1/p}\ge {\displaystyle \sum _{i=1}^n{\left({\displaystyle {\int }_a^b{\left(x_i^p\left(t\right)\right)}^{\prime }f\left(t\right)dt}\right)}^{1/p}}.$$

If $f$ is a nonincreasing function and $x_i(a)=0$ for all $i=1,\dots ,n$, then the reverse inequality is valid.

Thomson noted that (in the line) the Hausdorff measures can be considered to be fine variations for appropriate choices of derivation basis and set function. We show that this point of view remains interesting in a general separable metric space. Use of the “centered ball” basis yields an alternate description of the covering measures of Saint Raymond and Tricot. Use of a “closed set” basis yields the Ha usdorff measures. This paper may be considered a counterpart of [7], where the corresponding study of the packing measure may be found.

We prove that the smallest cardinality of a family $\mathcal{F}$ of real functions for which there is no non-zero function $g:ℝ\to ℝ$ with the property that $f\cdot g$ is almost continuous (connected, Darboux function, respectively) for all $f\in \mathcal{F}$, is equal to the cofinality of the continuum.

We show that the Scorza Dragoni theorem does not hold for the density topology and we prove some special forms of this theorem.

In 1963 Vituškin, Ivanov and Melnikov constructed a compact set in the plane with positive linear measure which cannot be mapped, using a contraction onto a segment. Their paper is very concise and obscure. The question of the existence of such a set arose recently again. For this reason in this paper we construct this set again with a detailed proof.

We show that the “standard triple bubble” is the shortest way to enclose and separate three areas in ${ℝ}^2$, assuming that the enclosed regions and the exterior region are connected.

A function $f$ that is almost everywhere the derivative of a continuous $AC{G}^{*}$ function is Perron integrable; a proof of this is given by a direct construction of continuous major and minor functions for $f$.

We introduce the concept of a path integral which integrates the path derivatives and recovers the primitives. The path integral is an extension of the Henstock integral. Moreover, we introduce the $E$-strong Lusin condition. Using this new concept, we give a descriptive definition of a path integral and a monotone theorem of a path differentiable function.

If $f:ℝ\to ℝ$ is weakly connected and for every $x\in ℝ$ there is an $n_x$ with $f^{n_x}(x)=1$ then $f(1)=1$. The analogous result holds for any Darboux function $f$ for which the set of all $n_x$ is bounded above.

In the paper I simplify the proof of the proposition which was the main tool in the work by Bruckner, Ceder, and Keston [2] concerning Darboux functions in the first class of Baire.

We show that a typical continuous real function generates all analytic sets as image of $G_{\delta }$-sets and all Borel sets as injective images of $G_{\delta }$-sets.