We establish an upper bound theorem for the Hausdorff dimension of limsup sets. This theorem together with a theorem of extraction of subsequences of balls is used to prove the sharpness of certain lower bound estimates established via mass transference principles.

We construct a continuous function possessing an infinite derivative that cannot be integrated using the approximately continuous extension of the Denjoy-Khinchin integral introduced by Kubota.

If local forms of Littlewood’s three principles are stated as axioms for an ordered field, then each principle is equivalent to the completeness axiom.

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences that has been used to define several new concepts in recent articles due to Das, Pal, and Adhikary. We first consider a weaker notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions, as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called "quasi-Cauchy Lipschitz functions" is introduced following the line of investigations in Beer and Garrido (Topology Appl., 208 (2016), 1-9) and again several coincidence results are proved. The motivation behind such kind of Lipschitz functions is ascertained by the observation that every real-valued ward continuous function defined on a metric space can be uniformly approximated by real-valued quasi-Cauchy Lipschitz functions.

We give a complete characterization of all real-valued functions on the unit circle \(S^1\) that can be represented by integrating the spherical distance on \(S^1\) with respect to a signed measure or a probability measure.

It is well known that the set of points of continuity of a function $f\colon \mathbb R\to \mathbb R$ is a $G_\delta$ set, and that for every $F_\sigma$ set $D$ there is an $f$ such that the set of points of discontinuities of $f$ is $D$. In this paper we decompose $D$ into the union of two disjoint $F_\sigma$ sets $U$ and $V$, such that $f_{|V}$ is discontinuous on $V$, and we show that $V$ is maximal in the sense that there is an $f$ such that $f_{|\overline{D}}$ is continuous on $U$.

Milne-type inequalities for \(h\)-convex functions involving conformable operators are established. Additionally, new results are presented that generalize various known inequalities.

Hille's theorem is a powerful classical result in vector measure theory. It asserts that the application of a closed, unbounded linear operator commutes with strong/Bochner integration of functions taking values in a Banach space. This note shows that Hille's theorem also holds in the setting of complete locally convex spaces.

Asymptotic \(L_p\)-convergence, which resembles convergence in \(L_p\), was introduced to address a question in diffusive relaxation. This note aims to compare asymptotic \(L_p\)-convergence with convergence in measure and in weak \(L_p\) spaces. One of the results characterizes convergence in measure on finite measure spaces in terms of asymptotic \(L_p\)-convergence.

We simplify the proofs of Colwell's theorems on Blaschke products.

We show two elementary versions of the Change of Variable Theorem (for the unidimensional Riemann integral) that are not a particular case of the influential general version due to Hyman Kestelman. Examples are given.

There is an error in the beginning of the proof of [1, Theorem 2] on page 398 between lines 10 and 19. We correct that error in this note.