Adding one random real

1996 ◽  
Vol 61 (1) ◽  
pp. 80-90 ◽  
Author(s):  
Tomek Bartoszyński ◽  
Andrzej Rosłanowski ◽  
Saharon Shelah
Keyword(s):  
Measure Zero ◽  
Continuous Functions ◽  
Random Real ◽  
Cardinal Invariants

AbstractWe study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

Order continuous measures

1964 ◽  
Vol 60 (2) ◽  
pp. 205-207 ◽  
Author(s):  
G. T. Roberts
Keyword(s):  
Topological Space ◽  
Linear Form ◽  
Vector Lattice ◽  
Measure Zero ◽  
Continuous Functions ◽  
Order Convergence ◽  
Locally Compact ◽  
Compact Topological Space ◽  
Continuous Measures ◽  
Order Continuous

1. Objective. It is possible to define order convergence on the vector lattice of all continuous functions of compact support on a locally compact topological space. Every measure is a linear form on this vector lattice. The object of this paper is to prove that a measure is such that every set of the first category of Baire has measure zero if and only if the measure is a linear form which is continuous in the order convergence.

scholarly journals Generalized Differentiability of Continuous Functions

2020 ◽  
Vol 4 (4) ◽  
pp. 56
Author(s):  
Dimiter Prodanov
Keyword(s):  
Mathematical Models ◽  
Measure Zero ◽  
Continuous Functions ◽  
Primitive Function ◽  
Generalized Differentiability ◽  
Differentiable Functions ◽  
Maximal Modulus ◽  
Physical Phenomena ◽  
Broad Generalization ◽  
Differentiation Theorem

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

scholarly journals Quasicontinuous functions and the topology of uniform convergence on compacta

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 911-917
Author(s):  
Lubica Holá ◽  
Dusan Holý
Keyword(s):  
Topological Space ◽  
Uniform Convergence ◽  
Continuous Functions ◽  
Hausdorff Topological Space ◽  
Cardinal Invariants ◽  
Weight Density ◽  
Quasicontinuous Functions

Let X be a Hausdorff topological space, Q(X,R) be the space of all quasicontinuous functions on X with values in R and ?UC be the topology of uniform convergence on compacta. If X is hemicompact, then (Q(X,R), ?UC) is metrizable and thus many cardinal invariants, including weight, density and cellularity coincide on (Q(X,R), ?UC). We find further conditions on X under which these cardinal invariants coincide on (Q(X,R), ?UC) as well as characterizations of some cardinal invariants of (Q(X,R), ?UC). It is known that the weight of continuous functions (C(R,R), ?UC) is ?0. We will show that the weight of (Q(R,R), ?UC) is 2c.

On Continuous Functions which are Partially Differentiable Almost Everywhere

1969 ◽  
Vol 12 (5) ◽  
pp. 668-672
Author(s):  
L.V. Toralballa
Keyword(s):  
Lebesgue Measure ◽  
Surface Area ◽  
Measure Zero ◽  
Continuous Functions ◽  
Lebesgue Measure Zero ◽  
Almost Everywhere

In the theory of surface area one meets situations where a function z = f(x, y) which is defined and continuous on a closed rectangle E, is partially differentiable on E except on a subset of E of Lebesgue measure zero.

scholarly journals Existence of Continuous Functions That Are One-to-One Almost Everywhere

2016 ◽  
Vol 118 (2) ◽  
pp. 269 ◽  
Author(s):  
Alexander J. Izzo
Keyword(s):  
Metric Space ◽  
Borel Measure ◽  
Measure Zero ◽  
Continuous Functions ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
One To One

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

Almost Somewhat Near Continuity and Near Regularity

2021 ◽  
Vol 7 (1) ◽  
pp. 88-99
Author(s):  
Zanyar A. Ameen
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
One To One ◽  
Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

Sign in / Sign up

Export Citation Format

Share Document