Non-constant Continuous Function Research Articles

The Nehari manifold approach for singular equations involving the p(x)-Laplace operator

We study the following singular problem involving the p(x)-Laplace operator , where is a nonconstant continuous function, Here, Ω is a bounded domain in with -boundary, λ is a positive parameter, are positive weight functions with compact support in Ω, and satisfy certain hypotheses () and (). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem .

Dynamics on sensitive and equicontinuous functions

The notions of sensitive and equicontinuous functions under semigroup action are introduced and intensively studied. We show that a transitive system is sensitive if and only if it has a sensitive pair if and only if it has a sensitive function. While there exists a minimal non-weakly mixing system such that every non-constant continuous function is sensitive, and a topological dynamical system is weakly mixing if and only if it is sensitive consistently with respect to (at least) any two non-constant continuous functions. We also get a dichotomy result for minimal systems -- every continuous function is either sensitive or equicontinuous.

Commuting pairs of self-adjoint elements in C *-algebras

Abstract We give a condition on commutativity of a pair of self-adjoint elements in a C *-algebra with respect to the continuous functional calculus. We also give an answer to the question raised by Jeang and Ko that if a non-constant continuous function totally spans the given C *-algebra.

Every locally connected functionally Hausdorff space is c-resolvable

For a cardinal κ, we say that a T1-space Y is a κ-retodic space if Y is partitioned by a family {Dξ}ξ∈κ of dense subsets such that the complement of every Dξ is totally disconnected. It is shown that “A locally connected space X is κ-resolvable if for every connected open subset U⊆X, there exists a κ-retodic space YU and a non-constant continuous function f:U⟶YU”. Consequently, every locally connected functionally Hausdorff space is c-resolvable, which is an answer to the question of K. Padmavally about resolvability of locally connected spaces, and also every c-irresolvable locally connected T1 space contains an open connected subset in which all continuous functions on it to any c-retodic space (all real continuous functions on it) are constant.

Nonconstant Continuous Functions whose Tangential Derivative Vanishes along a Smooth Curve

AbstractWe provide a simple example showing that the tangential derivative of a continuous function ϕ can vanish everywhere along a curve while the variation of ϕ along this curve is nonzero. We give additional regularity conditions on the curve and/or the function that prevent this from happening.

An elementary solution to the mountain climbers' problem

In this paper the following well known result is proved in a constructive elementary way: if f 1, f 2 : [0, 1] → [0,1] are locally non-constant continuous functions with f 1 (0) = f 2 (0) = 0, f 1 (1) = f 2 (1) = 1, then there are functions g 1, g 2 with the same properties such that $ f_1\circ g_1=f_2\circ g_2 $ .

A Non-Constant Continuous Function on the Plane Whose Integral on Every Line Is Zero

for every (doubly infinite) line l, where s denotes length measure, then f must be identically zero. This has been known for many years (see [5] for references), but only comparatively recently was it shown that this result fails in the absence of the global integrability condition. In fact, Zalcman [S; pp. 243-244], using a theorem of Arakelian [1; p. 1189] concerning tangential holomorphic approximation on unbounded sets, constructed a non-constant entire function t satisfying (1) on every line 1. The purpose of this note is to give a construction, similar to Zalcman's, but using only elementary complex analysis. As a bonus all the derivatives of the function we produce also have vanishing integrals on all lines. The main idea is to construct a non-constant entire function g whose derivatives satisfy

A Non-Constant Continuous Function on the Plane Whose Integral on Every Line Is Zero

A Non-Constant Continuous Function on the Plane Whose Integral on Every Line Is Zero

Distributions determined by conditioning on a single order statistic

LetX 1,X 2, …,X n(n ⩾ 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), −∞ ⩽a <b ⩽ ∞, the support ofF andX 1:n ⩽X 2:n ⩽ … ⩽X n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a +) andE {g(X)}. Then for some positive integers, 1 <s ⩽n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and $$F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)$$ . This leads to characterization of several distribution functions. A general form of this result is also stated.

Multiplicativity of the uniform norm and independent functions

It has long been known that there is a close connection between stochastic independence of continuous functions on an interval and space-filling curves [9]. In fact any two nonconstant continuous functions on [0, 1] which are independent relative to Lebesgue measure are the coordinate functions of a space filling curve. (The results of Steinhaus [9] have apparently been overlooked in more recent work in this area [3, 5, 6].)

Perturbation of functions by the paths of a Lévy process

In a recent paper Mountford [4] showed, using an ingenious probabilistic argument, that if X is a real-valued stable process with index α &lt; 1 and f: [0, ∞) → ℝ is a non-constant continuous function, thenwhere we use the notation |A| for the Lebesgue measure of a Lebesgue measurable set A ⊂ ℝ. The argument in [4] appears to make strong use of the strict scaling properties of X and the ‘intermediate value’ property of f.

Dispersion points and continuous functions

The aim of this paper is to answer the following question raised by J. Cobb and W. Voxman in 1980: If X is a connected space with a dispersion point p, and if ƒ: X→ X is a nonconstant continuous function, then is ƒ( p)= p? The answer to this question is negative, and we give a counterexample along with three theorems on a space with a dispersion point.

Invariant subspaces of Toeplitz operators

This paper is devoted to the problem of the existence of invariant subspaces for Toeplitz operators. Let Γ be a Lipschitzian arc in the plane and let f be a non-constant continuous functions on the unit circumference. It is proved that if there exists an open circle such that and if the modulus of continuity ωf of the function f satisfies the condition then the Toeplitz operator Tf in the Hardy space H2 has a nontrivial hyperinvariant subspace. For the proof of this theorem one makes use of the Lyubich-Matsaev theorem.

On The Generalized Cauchy Equation

It is the purpose of this note to prove the following theorem: Let ƒ: G → R be a non-constant continuous function with G a locally compact connected topological group and with R the real numbers. Let C = ƒ(G) and suppose that F: C × C → C is a junction such thatThen ƒ is monotone and open and F is continuous.

On the foundations of the calcul fonctionnel of Fréchet

In his thesis, t Frechet gave a very beautiful generalization of the theory of point sets and of the theory of real valued functions of a real variable. His functions are real valued but the range of the independent variable is an abstract class -S of elements q . He secures his results principally through the medium of a properly conditioned distance function 8, a generalization of the distance between two points, which associates with each pair q1 q2 of elements a real non-negative number 8 (qi q2 ) . He is thus enabled to secure the more important theorems of point set theory and of real function theory, especially those relating to continuous functions and their properties. This theory of Frechet has excited considerable interest and has received much attention from mathematicians. Various contributions to its foundations and to its content have been made. In the present paper we follow the example of Frechet in assuming once for all that 8 (qq) = 0 and that 8 (qi q2 ) = 8 ( q2 q ) . In other words we assume that the distance from an element to itself is zero and that the distance between two elements is independent of the order in which they are taken. In the first part of the paper we give very simple conditions on systems ( ; 8) which are sufficient for many purposes and which, in the case of compact sets, we show to be equivalent, so far as limit of a sequence is concerned, to the voisinage and thus to the e'cart of Frechet.t The remainder of the paper is devoted to the theory of functions on the sets Z of systems (Z; 8). In terms of the conditions already mentioned we generalize the results of Frechet and Hahn as to the existence of non-constant continuous functions. We give two very