相关论文: Possibly every real function is continuous on a no…

Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element…

We survey our recent result that for every continuous function there is an absolutely continuous homeomorphism such that the composition has a uniformly converging Fourier expansion. We mention the history of the problem, orginally stated…

It is shown that that for every Darboux function $F$ there is a non-constant continuous function $f$ such that $F+f$ is still Darboux. It is shown to be consistent --- the model used is iterated Sacks forcing --- that for every Darboux…

Consider the ring $C_c(X)_F$ of real valued functions which are discontinuous on a finite set with countable range. We discuss $(\mathcal{Z}_c)_F$-filters on $X$ and $(\mathcal{Z}_c)_F$-ideals of $C_c(X)_F$. We establish an analogous…

In this paper we show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function $F:I^2\to I$ is…

We prove that every weak solution to a certain class of infinitely degenerate quasilinear equations is continuous. An essential feature of the operators we consider is that their Fefferman-Phong associated metric may be non doubling with…

In this paper we prove a conjecture of J. Andrade, S. J. Miller, K. Pratt and M. Trinh, showing the existence of a non trivial infinite $F$-set over $\mathbb F_q[x]$ for every fixed $q$. We also provide the proof of a refinement of the…

Consider a Henselian rank one valued field $K$ of equicharacteristic zero along with the language $\mathcal{L}^{P}$ of Denef--Pas. Let $f: A \to K$ be an $\mathcal{L}^{P}$-definable (with parameters) function on a subset $A$ of $K^{n}$. We…

Answering a question asked by K.C. Ciesielski and T. Glatzer in 2013, we construct a $C^1$-smooth function $f$ on $[0,1]$ and a set $M \subset \operatorname{graph} f$ nowhere dense in $\operatorname{graph} f$ such that there does not exist…

We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving…

Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we…

The Fourier series of continuous functions of constant absolute value have interesting properties : according to the main theorems of the article, if the coefficients with positive indexes are square-summable with respect to a certain…

We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…

In this work we prove that an entire function $f(z)$ has only negative zeros if and only if its order is strictly less $1$, its root sequence is real-part dominating and there exists an nonnegative integer $m$ the real function…

It is well known that in Zermelo-Fraenkel (ZF) set theory any finite set is decidable. In this paper we discuss an extension of ZF where this result is no longer valid. Such an extension is quasi-set theory and it has its origin on problems…

Let $\mathbb{K}$ be an uncountable field of characteristic zero and let $f$ be a function from $\mathbb{K}^n$ to $\mathbb{K}$. We show that if the restriction of $f$ to every affine plane $L\subset\mathbb{K}^n$ is regular, then $f$ is a…

In this article we will investigate nonmeasurability with respect to some $\sigma$-ideals in Polish space $X,$ of images of subsets of $X$ by selected mappings defined on the space $X$. Among of them we answer the following question: "It is…

We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a…

Recently, in Axioms 10(2): 119 (2021), a nonclassical first-order theory T of sets and functions has been introduced as the collection of axioms we have to accept if we want a foundational theory for (all of) mathematics that is not weaker…

We provide answers to a question brought up by Erd\H{o}s about the construction of Wetzel families in the absence of the continuum hypothesis - a Wetzel family is a family $\mathcal{F}$ of entire functions on the complex plane which…