We provide a characterization of countably n-rectifiable measures in terms of $\sigma$-finiteness of the integral Menger curvature. We also prove that a finiteness condition on pointwise Menger curvature can characterize rectifiability of Radon measures. Motivated by the partial converse of Meurer's work by Kolasiński we prove that under suitable density assumptions there is a comparability between pointwise-Menger curvature and the sum over scales of the centered $\beta$-numbers at a point.

For classes of functions defined by $k$-th moduli of continuity, calculated in the norm of the symmetric space $X$, an estimate of the difference of two members of the sequence of operators of Riemann sums on a set of large measure is given. The estimate is given in terms of a fundamental function of the space $X$. Using this result, for a function $f$ from a given symmetric space, sufficient conditions are given for the almost everywhere convergence of the sequence of operators of Riemann sums to the Lebesgue integral of a given function.

There are continuous functions with complicated yet interesting sets of non-isolated non-strictly monotone points. This paper aims to characterize the sets of isolated and non-isolated non-strictly monotone points of the composition of continuous functions. Consequently, an uncountable dense set of measure zero in the real line and whose complement is also uncountable and dense is obtained.

The problem of characterizing continuous functions belonging to each $\mathcal A$-ring containing a fixed Darboux function is fully solved for a compact interval domain and solved with some minor exception for a real line domain. Many properties of generated $\mathcal A$-rings and their continuous functions are stated.

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of a real and finite square matrix $A$ (and for complex ones also). The elementary method developed avoids Jordan canonical form, eigenvectors, resolution of any linear system, matrix inversion, polynomial interpolation, complex integration, functional analysis, and generalized Fibonacci sequences. The basic tools are the Cayley-Hamilton theorem and the method of partial fraction decomposition. Two examples are given. We also show that such method applies to algebraic operators on infinite dimensional real Banach spaces.

Let $\Omega$ be an open subset of $\mathbb R^n (n \ge 2)$, and let $F : \Omega \rightarrow \mathbb R^n$ be a continuously differentiable map with countably many critical points. We show that $F$ is an open map. Let $G :\mathbb R^n \rightarrow \mathbb R^n (n \ge 1)$ be a continuously differentiable map such that $G(x) \rightarrow \infty$ as $x \rightarrow \infty$. Then it is proved that $G$ is surjective if and only if each connected component of the complement of the set of critical values of $G$ contains at least one image of $G$. Several applications of both theorems especially to complex analysis are presented.

Let $BV_p[0,1], 1 \le p \lt \infty$, be the Banach algebra of functions of bounded $p$-variation in the sense of Wiener. Recently, Kowalczyk and Turowska [9] proved that the multiplication in $BV_1[0,1]$ is an open bilinear mapping. We extend this result for all values of $p \in [1,\infty)$.

The Minkowski sum and Minkowski product can be considered as the addition and multiplication of subsets of $\mathbb R$. If we consider a compact subset $K \subseteq [0,1]$ and a power series $f$ which is absolutely convergent on $[0, 1]$, then we may use these operations and the natural topology of the space of compact sets to substitute the compact set $K$ into the power series $f$. Changhao Chen studied this kind of substitution in the special case of polynomials and showed that if we substitute the typical compact set $K \subseteq [0,1]$ into a polynomial, we get a set of Hausdorff dimension 0. We generalize this result and show that the situation is the same for power series where the coefficients converge to zero quickly. On the other hand we also show a large class of power series where the result of the substitution has Hausdorff dimension one.

We prove that if a set is ‘large’ in the sense of Erdős, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap length $\Delta$ of the progression, we improve a previous result of $o(\Delta)$ to $O(\Delta^\alpha)$ for any $\alpha \in (0,1)$. This improvement comes from a new approach relying on an iterative application of Szemerédi's Theorem.

A kind of Pettis integral representation for a Banach valued Itô process is given and its drift term is modified using a Girsanov Theorem.

Consider a real valued function defined, but not differentiable at some point. We use sequences approaching the point of interest to define and study sequential concepts of secant and cord derivatives of the function at the point of interest. If the function is the celebrated Weierstrass function, it follows from some of our results that the set cord derivates at any point coincides with the extended real line.

We resurrect an old definition of the linear measure of a metric continuum in terms of Steiner trees, independently due to Menger (1930) and Choquet (1938). We generalise it to any metric space and provide a proof of a little-known theorem of Choquet that it coincides with the outer linear measure for any connected metric space. As corollaries we obtain simple proofs of Gołąb’s theorem (1928) on the lower semicontinuity of linear measure of continua and a theorem of Bognár (1989) on the linear measure of the closure of a set. We do not use any measure theory apart from the definition of outer linear measure.

We characterize the primitives of Riemann integrable functions using the concept of strong differentiability. We also introduce the notion of essentially Riemann integrable function and characterize it descriptively using its primitive. We also give a characterization of this function that parallels Lebesgue's criterion for Riemann integrability.

We define variational measure with respect to measurable gauges and explore its properties. Further we establish that for continuous additive interval functions, it agrees with the usual variational measure on measurable sets.

Let $p, q$ be given positive numbers and $a, b$ non-negative ones. In this paper we study and characterize the class $\mathcal S(a,b,p,q)$ of all admissible ideals $\mathcal I \subset 2^\mathbb N$ with the following property $$\sum_{n \in \mathbb N} n^a a^p_n \lt \infty\quad \Rightarrow \quad \mathcal I − \lim n^ba^q_n = 0,$$ for all sequences $(a_n)$ of positive real numbers. In a series of corollaries we discuss special cases including, also, several previously published theorems on this topic.