Linearly continuous maps discontinuous on the graphs of twice differentiable functions

A function g : R n → R is linearly continuous provided its restriction g ` to every straight line ` ⊂ R n is continuous. It is known that the set D(g) of points of discontinuity of any linearly continuous g : R n → R is a countable union of isometric copies of (the graphs of) f P, where f : R n−1 → R is Lipschitz and P ⊂ R n−1 is compact nowhere dense. On the other hand, for every twice continuously differentiable function f : R → R and every nowhere dense perfect P ⊂ R there is a linearly continuous g : R 2 → R with D(g) = f P. The goal of this paper is to show that this last statement fails, if we do not assume that f 00 is continuous. More specifically, we show that this failure occurs for every continuously differentiable function f : R → R with nowhere monotone derivative, which includes twice differentiable functions f with such property. This generalizes a recent result of professor Ludek Zajicek and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasFALSEMinisterio de Ciencia e Innovación (MICINN)/FEDERunpu

Almost continuous Sierpinski-Zygmund functions under different set-theoretical assumptions

A function f : R → R is: almost continuous in the sense of Stallings, f ∈ AC, if each open set G ⊂ R2 containing the graph of f contains also the graph of a continuous function g : R → R; Sierpiński-Zygmund, f ∈ SZ (or, more generally, f ∈ SZ(Bor)), provided its restriction f M is discontinuous (not Borel, respectively) for any M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński-Zygmund almost continuous function f : R → R (i.e., an f ∈ SZ ∩ AC) cannot be constructed in ZFC; however, an f ∈ SZ ∩ AC exists under the additional set-theoretical assumption cov(M) = c, that is, that R cannot be covered by less than c-many meager sets. The primary purpose of this paper is to show that the existence of an f ∈ SZ∩AC is also consistent with ZFC plus the negation of cov(M) = c. More precisely, we show that it is consistent with ZFC+cov(M) < c (follows from the assumption that non(N ) < cov(N ) = c) that there is an f ∈ SZ(Bor)∩AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either cov(M) = c or non(N ) < cov(N ) = c, the lineability and the additivity coefficient of the class of all almost continuous Sierpiński-Zygmund functions. Several open problems are also stated

Additivity coefficients for all classes in the algebra of Darboux-Like maps on R

The class D of generalized continuous functions on R known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property. The algebra A(D) of classes of functions generated by these families contains 17 atoms. In this work we will calculate the values of the additivity coefficient A(F) for all atoms F in the algebra A(D). We also determine the values A(F) for a lot of other families F∈A(D). Open questions and new directions of research shall also be provided

Some properties of differentiable p-adic functions

In this paper, using the tools from the lineability theory, we distinguish certain subsets of p-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional algebraic structure: (i) continuously differentiable but not strictly differentiable functions, (ii) strictly differentiable functions of order r but not strictly differentiable of order r + 1, (iii) strictly differentiable functions with zero derivative that are not Lipschitzian of any order α > 1, (iv) differentiable functions with unbounded derivative, and (v) continuous functions that are differentiable on a full set with respect to the Haar measure but not differentiable on its complement having cardinality the continuum.Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasFALSEMinisterio de Ciencia e Innovación (MICINN)/FEDERINSFunpu

Classical vs. non-Archimedean analysis: an approach via algebraic genericity

In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem