‎INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO ‎‎COMPARABLE CONTRA-$\alpha-$CONTINUOUS (CONTRA-$C-$CONTINUOUS) FUNCTIONS

‎A necessary and sufficient condition in terms of lower cut sets ‎are given for the insertion of a contra-continuous function ‎between two comparable real-valued functions on such topological ‎spaces that kernel of sets are open‎.

Semi-smooth Points in Some Classical Function Spaces

Results in Mathematics ◽

10.1007/s00025-021-01545-9 ◽

2021 ◽

Vol 77 (1) ◽

Author(s):

Beata Derȩgowska ◽

Beata Gryszka ◽

Karol Gryszka ◽

Paweł Wójcik

Keyword(s):

Continuous Function ◽

Metric Space ◽

Continuous Functions ◽

Sufficient Condition ◽

Compact Metric Space ◽

Spaces Of Continuous Functions ◽

Classical Function ◽

Necessary And Sufficient ◽

Vector Valued

AbstractThe investigations of the smooth points in the spaces of continuous function were started by Banach in 1932 considering function space $$\mathcal {C}(\Omega )$$ C ( Ω ) . Singer and Sundaresan extended the result of Banach to the space of vector valued continuous functions $$\mathcal {C}(\mathcal {T},E)$$ C ( T , E ) , where $$\mathcal {T}$$ T is a compact metric space. The aim of this paper is to present a description of semi-smooth points in spaces of continuous functions $$\mathcal {C}_0(\mathcal {T},E)$$ C 0 ( T , E ) (instead of smooth points). Moreover, we also find necessary and sufficient condition for semi-smoothness in the general case.

On Weighted Polynomial Approximation With Gaps

Nagoya Mathematical Journal ◽

10.1017/s0027763000009119 ◽

2005 ◽

Vol 178 ◽

pp. 55-61 ◽

Cited By ~ 10

Author(s):

Guantie Deng

Keyword(s):

Banach Space ◽

Continuous Function ◽

Polynomial Approximation ◽

Continuous Functions ◽

Sufficient Condition ◽

Weighted Polynomial Approximation ◽

Weighted Banach Space ◽

Nonnegative Continuous Function ◽

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.

Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

Advances in Pure and Applied Mathematics ◽

10.1515/apam-2014-0053 ◽

2015 ◽

Vol 6 (3) ◽

Author(s):

Roman A. Veprintsev

Keyword(s):

Continuous Function ◽

Harmonic Analysis ◽

Unit Sphere ◽

Continuous Functions ◽

Sufficient Condition ◽

The Family ◽

AbstractWe establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊

Insertion of a Contra-continuous Function between Two Comparable Real-valued Functions

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.3120.2135 ◽

2019 ◽

pp. 21-35 ◽

Cited By ~ 1

Author(s):

Majid Mirmiran ◽

Binesh Naderi

Keyword(s):

Continuous Function ◽

Topological Spaces ◽

Sufficient Condition ◽

A necessary and sufficient condition in terms of lower cut sets are given for the insertion of a contra-continuous function between two comparable real-valued functions on such topological spaces that kernel of sets are open.

Ideal convergence of nets of functions with values in uniform spaces

Filomat ◽

10.2298/fil1720281m ◽

2017 ◽

Vol 31 (20) ◽

pp. 6281-6292

Author(s):

Athanasios Megaritis

Keyword(s):

Continuous Function ◽

Topological Space ◽

Uniform Space ◽

Continuous Functions ◽

Sufficient Condition ◽

Compact Sets ◽

Uniform Spaces ◽

Ideal Convergence ◽

We consider the pointwise, uniform, quasi-uniform, and the almost uniform I-convergence for a net (fd)d?D of functions from a topological space X into a uniform space (Y,U), where I is an ideal on D. The purpose of the present paper is to provide ideal versions of some classical results and to extend these to nets of functions with values in uniform spaces. In particular, we define the notion of I-equicontinuous family of functions on which pointwise and uniform I-convergence coincide on compact sets. Generalizing the theorem of Arzel?, we give a necessary and sufficient condition for a net of continuous functions from a compact space into a uniform space to I-converge pointwise to a continuous function.

XV.—On a Test for Continuity.

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600011676 ◽

1908 ◽

Vol 28 ◽

pp. 249-258

Author(s):

W. H. Young

Keyword(s):

Uniform Convergence ◽

Continuous Functions ◽

Usual Method ◽

Sufficient Condition ◽

§ 1. THE usual method of proving that a function defined as the limit of a sequence of continuous functions is continuous is by proving that the convergence is uniform. This method may fail owing to the presence of points at which the convergence is non-uniform although the limiting function is continuous. In such a case it would be necessary to apply a further test, e.g. that of Arzelà (“uniform convergence by segments”).In some cases the continuity may be proved directly by means of a totally different principle, without reference to modes of convergence at all. It is, in fact, a necessary and sufficient condition for the continuity of a function that it should be possible to express it at the same time as the limit of a monotone ascending and of a monotone descending sequence of continuous functions.

On Global Inverse Theorems of Szász and Baskakov Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-027-2 ◽

1979 ◽

Vol 31 (2) ◽

pp. 255-263 ◽

Cited By ~ 15

Author(s):

Z. Ditzian

Keyword(s):

Rate Of Convergence ◽

Exponential Growth ◽

Polynomial Growth ◽

Continuous Functions ◽

Sufficient Condition ◽

Baskakov Operators ◽

Inverse Theorems ◽

Image Position ◽

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

Metrisation of Moore spaces and abstract topological manifolds

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700031178 ◽

1997 ◽

Vol 56 (3) ◽

pp. 395-401 ◽

Cited By ~ 1

Author(s):

David L. Fearnley

Keyword(s):

Recent Work ◽

Topological Spaces ◽

Sufficient Condition ◽

Moore Space ◽

Global Coherence ◽

Topological Manifolds ◽

Necessary And Sufficient ◽

Moore Spaces ◽

Significant Class

The problem of metrising abstract topological spaces constitutes one of the major themes of topology. Since, for each new significant class of topological spaces this question arises, the problem is always current. One of the famous metrisation problems is the Normal Moore Space Conjecture. It is known from relatively recent work that one must add special conditions in order to be able to get affirmative results for this problem. In this paper we establish such special conditions. Since these conditions are characterised by local simplicity and global coherence they are referred to in this paper generically as “abstract topological manifolds.” In particular we establish a generalisation of a classical development of Bing, giving a proof which is complete in itself, not depending on the result or arguments of Bing. In addition we show that the spaces recently developed by Collins designated as “W satisfying open G(N)” are metrisable if they are locally separable and locally connected and regular. Finally, we establish a new necessary and sufficient condition for spaces to be metrisable.

Oscillation and Global Attractivity in a Periodic Delay Equation

Canadian Mathematical Bulletin ◽

10.4153/cmb-1996-035-9 ◽

1996 ◽

Vol 39 (3) ◽

pp. 275-283 ◽

Cited By ~ 23

Author(s):

J. R. Graef ◽

C. Qian ◽

P. W. Spikes

Keyword(s):

Global Attractivity ◽

Positive Periodic Solution ◽

Continuous Functions ◽

Sufficient Condition ◽

All Solutions ◽

Periodic Delay ◽

Image Position ◽

Delay Differential ◽

AbstractConsider the delay differential equationwhere α(t) and β(t) are positive, periodic, and continuous functions with period w > 0, and m is a nonnegative integer. We show that this equation has a positive periodic solution x*(t) with period w. We also establish a necessary and sufficient condition for every solution of the equation to oscillate about x*(t) and a sufficient condition for x*(t) to be a global attractor of all solutions of the equation.

Proximinal subspaces ofA(K)of finite codimension

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203212308 ◽

2003 ◽

Vol 2003 (39) ◽

pp. 2501-2505

Author(s):

T. S. S. R. K. Rao

Keyword(s):

Compact Subset ◽

Convex Set ◽

Compact Convex ◽

Continuous Functions ◽

Finite Codimension ◽

Sufficient Condition ◽

Choquet Simplex ◽

Compact Convex Set ◽

We study an analogue of Garkavi's result on proximinal subspaces ofC(X)of finite codimension in the context of the spaceA(K)of affine continuous functions on a compact convex setK. We give an example to show that a simple-minded analogue of Garkavi's result fails for these spaces. WhenKis a metrizable Choquet simplex, we give a necessary and sufficient condition for a boundary measure to attain its norm onA(K). We also exhibit proximinal subspaces of finite codimension ofA(K)when the measures are supported on a compact subset of the extreme boundary.