Related papers: An affirmative answer to a problem of Cater

Let A be an arbitrary countable set of reals, for example A=Q. Let g be an arbitrary mapping from A into the positive reals, for example g(a)=2^a. We show how a strictly increasing real function f can be constructed such that f'(x)=g(x) for…

In this work we study the following classical still challenging Calculus problem: {\it If $f:(0,\infty)\to\mathbb{R}$ is a continuous function, for which the sequence $\{f(nx)\}$ tends to zero, for every positive $x$, as $n$ tends to…

We give a short proof, that can be used in an introductory real analysis course, that if a function that is defined on the set of real numbers is continuous on a countable dense set, then it is continuous on an uncountable set. This is done…

Since the theory developed by Georg Cantor, mathematicians have taken a sharp interest in the sizes of infinite sets. We know that the set of integers is infinitely countable and that its cardinality is Aleph0. Cantor proved in 1891 with…

A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that…

Let $f(n)$ count the number of subsets of $\{1,...,n\}$ without an element dividing another. In this paper I show that $f(n)$ grows like the $n$-th power of some real number, in the sense that $\lim_{n\rightarrow \infty}f(n)^{1/n}$ exists.…

Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each…

For a function $f\colon [0,1]\to\mathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $f\colon [0,1]\to\mathbb R$ and a Cantor set $D\subset [0,1]$ with $\{0,1\}\subset D$, we obtain conditions equivalent…

We denote the local ``little" Lipschitz constant of a function $f: {{\mathbb R}}\to { {\mathbb R}}$ by $ {\mathrm{lip}}f$. In this paper we settle the following question: For which sets $E {\subset} { {\mathbb R}}$ is it possible to find a…

It is shown that for a function $f:\mathbb R^2\to \mathbb R$ which is measurable with respect to the first variable and upper semicontinuous quasicontinuous and increasing with respect to the second variable there exists a Caratheodory's…

This article explores the model-dependent nature of set cardinality, emphasizing that cardinality is not absolute but varies across different axiomatic frameworks. Although Cantor's diagonal argument shows the real numbers are…

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…

Erd\H{o}s \cite{MR168482} proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal{F}$ of (real or complex) analytic functions, such that $\big\{ f(x) \ : \ f \in \mathcal{F} \big\}$ is…

HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert's proof of transcendence of…

We study the problem of the existence of increasing and continuous solutions $\varphi\colon[0,1]\to[0,1]$ such that $\varphi(0)=0$ and $\varphi(1)=1$ of the functional equation \begin{equation*}…

A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the…

Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight aleph_1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add…

We construct an iterated function system consisting of strictly increasing contractions $f,g\colon [0,1]\to [0,1]$ with $f([0,1])\cap g([0,1])=\emptyset$ and such that its attractor has positive Lebesgue measure.

We consider approximations of a continuous function on a countable normed Fr\'{e}chet space by analytic and $*$-analytic. Also we found a criterium of the existence of an extension of a continuous function from a dense subspace of a…

We construct a separate continuous function $f\colon\mathbb{Q}\times\mathbb{Q}\to[0,1] $ and a dense subset $D\subseteq \mathbb{Q}\times\mathbb{Q}$ such that $f[D]$ is not dense in $f[\mathbb{Q}\times\mathbb{Q}]$, in other words, $f$ is…