We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof doesn’t require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weake...

In this paper, we first study a nonsmooth steepest descent method for nonsmooth functions defined on a Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitz continuity. As an application, the established algorithm is used to search for the minimizer of a...

tive continuity is a distributed invariant relationship between parts of flowing shapes. The l techniques presented here were developed for making the behavioral dynamics of x processes more recognizable, but are equally applicable to assisting in the recognition of in images. Regularizing a sequence using a constraint of derivative continuity is lent to using a bimodal smoothing kernel, produc...

In this paper, we derive a generalized multiplicative Hardy-LittlewoodPolya type inequality, as well as several related additive inequalities, for functions of operators in Hilbert spaces. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of the operator. We then apply the results to solve the following p...

The purpose of this article is to establish Jackson-type inequality in the polydiscs UN of C for holomorphic spaces X, such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely, E k(f,X) (−→ 1/k, f,X ) , where E k(f,X) is the deviation of the best approximation of f ∈ X by polynomials of degree at most kj about the jth variable zj with respect to the X-metric and ...

|ξ| ≤ f(x, u, ξ) ≤ L(1 + |ξ|). A function u ∈ W 1,p loc (Ω) is a local minimizer of F in Ω if F (u; spt (v − u)) ≤ F (v; spt (v − u)) , for every v ∈ W 1,p loc (Ω) such that spt (v − u) ⊂⊂ Ω. Well known results due to Giaquinta and Giusti [13, 15] ensure that local minimizers of F are locally α-Hölder continuous for some α < 1. According to Meyers’ example in [19], when f is not continuous in Ω...

The recently developed technique of Bohrification associates to a (unital) C*-algebra A 1. the Kripke model, a presheaf topos, of its classical contexts; 2. in this Kripke model a commutative C*-algebra, called the Bohrification of A; 3. the spectrum of the Bohrification as a locale internal in the Kripke model. We propose this locale, the ‘state space’, as a (n intuitionistic) logic of the phy...

and Applied Analysis 3 that the function f : (X, α) → (Y, β) is strongly proximally continuous on A if ∀E ⊂ A, ∀S ⊂ X, EαS 󳨐⇒ f (E) βf (S) . (5) Finally, one says that f is strongly proximally continuous on B if f is strongly proximally continuous on A, for every A ∈ B. We shall use the notation C B (X, Y) to denote the family of the functions from X to Y which are strongly proximally continuou...

it is well known that a microperiodic function mapping a topological group into reals, which is continuous at some point is constant. we introduce the notion of a microperiodic multifunction, defined on a topological group with values in a metric space, and study regularity conditions implying an analogous result. we deal with vietoris and hausdorff continuity concepts.stability of microperiodi...

In this paper we investigate the continuity of the Drazin inverse of a bounded linear operator on Banach space. Then as a corollary, among other things, we get the well known result of Campbell and Meyer ([1]) for the continuity of the Drazin inverse of square matrix.