Related papers: A separate continuous function not feebly continuo…

We provide the following result and its discrete equivalent: Let $f \colon I^n \to \mathbb{R}^{n-1}$ be a continuous function. Then, there exist a point $p \in \mathbb{R}^{n-1}$ and a compact subset $S \subset…

We prove that every continuous function $f:E\to Y$ depends on countably many coordinates, if $E$ is an $(\aleph_1,\aleph_0)$-invariant pseudo-$\aleph_1$-compact subspace of a product of topological spaces and $Y$ is a space with a regular…

We address a problem posed in [1] by demonstrating through an example that, in the absence of separability, the property of sequential cone compactness does not generally imply cone compactness.

Abstrct: In this note, by considering fractionally linear functions over a finite field and consequently developing an abstract sequence, we study some of its properties.

In this paper we study the self-similar processes with stationary increments in a discrete-time setting. Different from the continuous-time case, it is shown that the scaling function of such a process may not take the form of a power…

We construct a finitely presented group $G$ with non-quadratic Dehn function $f$ majorizable by a quadratic function on arbitrary long intervals.

In this paper we continue our research of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. In particular, we prove that for a metrizable topological space $X$, a…

A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel…

We consider continuous functions f : [0,1] \to R that cut the real axis at every point of a measurable set of positive measure and we construct examples where f fails to have bounded variation, and at the opposite end, where f admits…

We give a (consistent) example of a first-countable continuum that is not a remainder of the real line.

The separately continuity topology is considered and some its properties are investigated. With help of these properties a generalization of Sierpinski theorem on determination of real separately continuous function by its values on an…

We construct several new classes of transcendental entire functions, f, such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider's web. We show that some of these classes have a degree of…

We consider the discrete time stopping problem \[ V(t,x) = \sup_{\tau}E_{(t,x)}[g(\tau, X_\tau)],\] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set…

We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…

In a series of papers, M.Talagrand, the second author and others investigated at length the properties and structure of pointwise compact sets of measurable functions. A number of problems, interesting in themselves and important for the…

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf…

Let $M$ be a closed smooth manifold and let $f:M\to M$ be a diffeomorphism. $C^1$-generically, a continuum-wise expansive satisfies Axiom A without cycles. Moreover, there is a partially hyperbolic diffeomorphism $f$ such that it is not…

We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an…

Does there exist an increasing absolutely continuous function, $f: [0,1] \rightarrow \mathbb R$ such that $\{x: f'(x)=0\}$ is both countable and dense? This problem was proposed by F.S. Cater about two decades ago. We give an affirmative…

A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be…