scholarly journals A uniformly continuous function on [0,1] that is everywhere different from its infimum

1984 ◽  
Vol 111 (2) ◽  
pp. 333-340 ◽  
Author(s):  
William Julian ◽  
Fred Richman
Keyword(s):  
Continuous Function ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

On the Extension of Uniformly Continuous Functions(1)

1975 ◽  
Vol 18 (1) ◽  
pp. 143-145 ◽  
Author(s):  
L. T. Gardner ◽  
P. Milnes
Keyword(s):  
Continuous Function ◽  
Uniform Space ◽  
Continuous Extension ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Totally Bounded ◽  
Special Cases ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

AbstractA theorem of M. Katětov asserts that a bounded uniformly continuous function f on a subspace Q of a uniform space P has a bounded uniformly continuous extension to all of P. In this note we give new proofs of two special cases of this theorem: (i) Q is totally bounded, and (ii) P is a locally compact group and Q is a subgroup, both P and Q having the left uniformity.

scholarly journals U(X) as a ring for metric spaces X

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1981-1984 ◽  
Author(s):  
Sànchez Cabello
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Infinite Subset ◽  
Short Paper ◽  
Uniformly Continuous Function ◽  
Uniformly Continuous

In this short paper, we will show that the space of real valued uniformly continuous functions defined on a metric space (X,d) is a ring if and only if every subset A ? X has one of the following properties: ? A is Bourbaki-bounded, i.e., every uniformly continuous function on X is bounded on A. ? A contains an infinite uniformly isolated subset, i.e., there exist ? > 0 and an infinite subset F ? A such that d(a,x) ? ? for every a ? F, x ? X n \{a}.

scholarly journals Convolutions with the Continuous Primitive Integral

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Erik Talvila
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Bounded Variation ◽  
Real Line ◽  
Fubini Theorem ◽  
Function Of Bounded Variation ◽  
Distributional Derivative ◽  
The Real ◽  
Integrable Distribution ◽  
Uniformly Continuous

IfFis a continuous function on the real line andf=F′is its distributional derivative, then the continuous primitive integral of distributionfis∫abf=F(b)−F(a). This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Under the Alexiewicz norm, the space of integrable distributions is a Banach space. We define the convolutionf∗g(x)=∫−∞∞f(x−y)g(y)dyforfan integrable distribution andga function of bounded variation or anL1function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. Forgof bounded variation,f∗gis uniformly continuous and we have the estimate‖f∗g‖∞≤‖f‖‖g‖ℬ𝒱, where‖f‖=supI|∫If|is the Alexiewicz norm. This supremum is taken over all intervalsI⊂ℝ. Wheng∈L1, the estimate is‖f∗g‖≤‖f‖‖g‖1. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

scholarly journals Upward and downward statistical continuities

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2265-2273 ◽  
Author(s):  
Hüseyin Çakallı
Keyword(s):  
Continuous Function ◽  
Bounded Subset ◽  
Real Numbers ◽  
Cauchy Sequences ◽  
Uniformly Continuous

A real valued function f defined on a subset E of R, the set of real numbers, is statistically upward (resp. downward) continuous if it preserves statistically upward (resp. downward) half quasi-Cauchy sequences; A subset E of R, is statistically upward (resp. downward) compact if any sequence of points in E has a statistically upward (resp. downward) half quasi-Cauchy subsequence, where a sequence (xn) of points in R is called statistically upward half quasi-Cauchy if lim n?? 1/n |{k ? n : xk- xk+1 ? ?}| = 0, and statistically downward half quasi-Cauchy if lim n??1/n |{k ? n : xk+1 - xk ? ?}| = 0 for every ? > 0. We investigate statistically upward and downward continuity, statistically upward and downward half compactness and prove interesting theorems. It turns out that any statistically upward continuous function on a below bounded subset of R is uniformly continuous, and any statistically downward continuous function on an above bounded subset of R is uniformly continuous.

scholarly journals Cyclicity in Dirichlet Spaces

2019 ◽  
Vol 62 (02) ◽  
pp. 247-257 ◽  
Author(s):  
Y. Elmadani ◽  
I. Labghail
Keyword(s):  
Continuous Function ◽  
Unit Circle ◽  
Borel Measure ◽  
Dirichlet Space ◽  
Sufficient Condition ◽  
Dirichlet Spaces ◽  
Closed Unit Disk ◽  
Finite Borel Measure ◽  
Weighted Dirichlet Space ◽  
Uniformly Continuous

AbstractLet $\unicode[STIX]{x1D707}$ be a positive finite Borel measure on the unit circle and ${\mathcal{D}}(\unicode[STIX]{x1D707})$ the associated harmonically weighted Dirichlet space. In this paper we show that for each closed subset $E$ of the unit circle with zero $c_{\unicode[STIX]{x1D707}}$ -capacity, there exists a function $f\in {\mathcal{D}}(\unicode[STIX]{x1D707})$ such that $f$ is cyclic (i.e., $\{pf:p\text{ is a polynomial}\}$ is dense in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ ), $f$ vanishes on $E$ , and $f$ is uniformly continuous. Next, we provide a sufficient condition for a continuous function on the closed unit disk to be cyclic in ${\mathcal{D}}(\unicode[STIX]{x1D707})$ .

scholarly journals Polynomials and functions with finite spectra on locally compact Abelian groups

1995 ◽  
Vol 51 (1) ◽  
pp. 33-42 ◽  
Author(s):  
B. Basit ◽  
A.J. Pryde
Keyword(s):  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Finite Spectrum ◽  
Beurling Algebra ◽  
Uniformly Continuous Function ◽  
Compact Abelian Groups ◽  
Indefinite Integral ◽  
Beurling Algebras ◽  
Primary Ideals ◽  
Uniformly Continuous

In this paper we define polynomials on a locally compact Abelian group G and prove the equivalence of our definition with that of Domar. We explore the properties of polynomials and determine their spectra. We also characterise the primary ideals of certain Beurling algebras on the group of integers Z. This allows us to classify those elements of that have finite spectrum. If ϕ is a uniformly continuous function with bounded differences then there is a Beurling algebra naturally associated with ϕ. We give a condition on the spectrum of ϕ relative to this algebra which ensures that ϕ is bounded. Finally we give spectral conditions on a bounded function on ℝ that ensure that its indefinite integral is bounded.

Atsuji completions vis-à-vis hyperspaces

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Tanvi Jain ◽  
S. Kundu
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Metrizable Spaces ◽  
Uniformly Continuous ◽  
Hyperspace Topologies

AbstractA metric space (X, d) is called an Atsuji space if every real-valued continuous function on (X, d) is uniformly continuous. It is well-known that an Atsuji space must be complete. A metric space (X, d) is said to have an Atsuji completion if its completion ($$ \hat X $$, d) is an Atsuji space. In this paper, we study twelve equivalent (external) characterizations for a metric space to have an Atsuji completion in terms of hyperspace topologies. We also characterize topologically those metrizable spaces whose completions are Atsuji spaces.

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