Related papers: Effectively Open Real Functions

Given a function $f\colon \mathbb{R}\to \mathbb{R}$, the so-called "little lip" function $\operatorname{lip} f$ is defined as follows: \begin{equation*} \operatorname{lip} f(x)=\liminf_{r{\scriptscriptstyle \searrow} 0}\sup_{|x-y|\le r}…

We give necessary and sufficient conditions on a function $f:[0,1]\to {0,1,2,...,\omega,\continuum}$ under which there exists a continuous function $F:[0,1]\to [0,1]$ such that for every $y\in[0,1]$ we have $|F^{-1}(y)|=f(y)$.

We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as…

This paper provides a new and more direct proof of the assertion that a Turing computable function of the natural numbers is primitive recursive if and only if the time complexity of the corresponding Turing machine is bounded by a…

Let $X$ be an arbitrary set. Then a topology $t$ on $X$ is said to be completely useful if every upper semicontinuous linear (total) preorder $\precsim$ on $X$ can be represented by an upper semicontinuous real-valued order preserving…

In this note, we continue to highlight some applications of Theorem 1 of [3]. Here is a sample: Let $X$ be an open set in ${\bf C}^n$, $\Omega$ an open convex set in ${\bf C}$ and $f, g : X\to {\bf C}$ two holomorphic functions such that…

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is…

We study the ring of rational functions admitting a continuous extension to the real affine space. We establish several properties of this ring. In particular, we prove a strong Nullstelensatz. We study the scheme theoretic properties and…

We give an exposition of Natural Topology (NToP), which highlights its advantages for exact computation. The NToP-definition of the real numbers (and continuous real functions) matches recent expert recommendations for exact real…

Let $X$ be a Borel subset of the Cantor set \textbf{C} of additive or multiplicative class ${\alpha},$ and $f: X \to Y$ be a continuous function with compact preimages of points onto $Y \subset \textbf{C}.$ If the image $f(U)$ of every…

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U…

We present an extension to the $\mathtt{mathlib}$ library of the Lean theorem prover formalizing the foundations of computability theory. We use primitive recursive functions and partial recursive functions as the main objects of study, and…

We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J.…

A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative to…

Let S\subset (0,1). Given a known function f:S\to (0,1), we consider the problem of using independent tosses of a coin with probability of heads p (where p\in S is unknown) to simulate a coin with probability of heads f(p). We prove that if…

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…

In the article a technique of the usage of $f$-continuous functions (on mappings) and their families is developed. A proof of the Urysohn's Lemma for mappings is presented and a variant of the Brouwer-Tietze-Urysohn Extension Theorem for…

Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as…

A Cech closure space $(X,u)$ is a set $X$ with a (Cech) closure operator $u$ which need not be idempotent. Many properties which hold in topological spaces hold in Cech closure spaces as well. The notions of proper (splitting) and…