A topological characterisation of the implication 'differentiability implies continuity'

It is well known that differentiable functions defined on R are continuous. However, this result assumes that one uses the usual topology. In this paper, an example is given of a differentiable, nowhere continuous function by changing the basic open sets at just one point. And also a characterization is given of the implication ‘differentiability implies continuity’.     

The implicit function theorem in a non-Archimedean setting

In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from Nⁿ to N^m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from Nⁿ to Nⁿ and the implicit function theorem for functions from Nⁿ to N^m with m<n.     

Singularities of o‐minimal Peano derivatives

The notion of Peano differentiability generalizes the differentiability in the usual sense to higher order. Peano differentiable functions have derivatives which are sometimes differentiable or continuous or not even locally bounded. We give a complete characterisation of the sets in which Peano differentiable functions which are definable in an o‐minimal expansion of a real closed field are continuously differentiable. Thereby, we also distinguish between several kinds of discontinuities (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

Analysis of Symmetric Matrix Valued Functions

For any symmetric function f: ? ⁿ  → ? ⁿ , one can define a corresponding function on the space of n × n real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Fréchet differentiability, and continuous differentiability.     

Integral Representations,Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on R ⁿ. Our integral representation is an extension of Sobolev's integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.     

An upper gradient approach to weakly differentiable cochains

The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio–Kirchheim?s theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen–Koskela?s concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with p-integrable upper gradient in n-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey–Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.     

About differentiability of order one of quasiconvex functions onR n

This paper is devoted to the study of the different kinds of differentiability of quasiconvex functions onR ⁿ . For these functions, we show that Gâteaux-differentiability and Fréchet-differentiability are equivalent; we study the properties of the directional derivatives; and we show that if, for a quasiconvex function, the directional derivatives atx are all finite and two-sided, the function is differentiable atx.     

Convexity of trace functionals and Schrödinger operators

Let H be a semi-bounded self-adjoint operator on a separable Hilbert space. For a certain class of positive, continuous, decreasing, and convex functions F we show the convexity of trace functionals of the form tr(F(H+U−ε(U)))−ε(U), where U is a bounded, self-adjoint operator and ε(U) is a normalizing real function—the Fermi level—which may be identical zero. If additionally F is continuously differentiable, then the corresponding trace functional is Fréchet differentiable and there is an expression of its gradient in terms of the derivative of F. The proof of the differentiability of the trace functional is based upon Birman and Solomyak's theory of double Stieltjes operator integrals. If, in particular, H is a Schrödinger-type operator and U a real-valued function, then the gradient of the trace functional is the quantum mechanical expression of the particle density with respect to an equilibrium distribution function f=−F^′. Thus, the monotonicity of the particle density in its dependence on the potential U of Schrödinger's operator—which has been understood since the late 1980s—follows as a special case.     

On differentiability of mappings of finite-dimensional domains into Banach spaces

The well-known Stepanov criterion of the differentiability (approximate differentiability) of real functions is generalized to mappings of subsets ofR ⁿ into Banach spaces satisfying the Rieffel sharpness condition, in particular, reflexive Banach spaces. For Banach spaces that do not satisfy the Rieffel sharpness condition, this criterion is not true. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 3–11, January, 1999.     

A local mean value theorem for functions on non-archimedean field extensions of the real numbers

In this paper, we review the definition and properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. Then we define and study n-times locally uniform differentiable functions at a point or on a subset of N. In particular, we study the properties of twice locally uniformly differentiable functions and we formulate and prove a local mean value theorem for such functions.     

Best polynomial approximations in L 2 of classes of 2π-periodic functions and exact values of their widths

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity ofmth order in the space L ₂ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L ₂.     

On the convergence of quasi-newton methods for nonsmooth problems

We develop a theory of quasi-New ton and least-change update methods for solving systems of nonlinear equations F(x) = 0. In this theory, no differentiability conditions are necessary. Instead, we assume that Fcan be approximated, in a weak sense, by an affine function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-differentiable functions and to partially differentiable functions.     

The Dini condition and regularity of weak solutions of elliptic equations

Regularity results in domains of Euclidean n-space are established for generalized solutions of second order elliptic equations for which the coefficients of the differential operator and the nonhomogeneous term satisfy a Dini criterion. Generalized solutions are shown to be essentially classical solutions and a bound for the modulus of continuity of second order partial derivatives of the solution is established which yields Weyl's lemma as a corollary. A differentiability theorem is also established for the case the terms of the equation have further differentiability properties.     

Fractional non-Archimedean calculus in one variable

Let ?? be a natural number. A function f: ? _p ?? K into a non-Archimedeanly valued complete field K ? ? _p is ??-times continuously differentiable if and only if its Mahler coefficients (a _n ) _n??? obey |a _n |n ^?? ?? 0 as n ?? ??. For a real number r ?? 0, this suggests the ad hoc definition by [1] of a C ^r -function f: ? _p ?? K by asking its Mahler coefficients (a _n ) _n??? to satisfy |a _n |n ^r ?? 0 as n?? ??. We will present for functions f: X ?? K on subsets X ? K without isolated points a general pointwise notion of r-fold differentiability through iterated difference quotients, subsequently shown on the domain X = ? _p to coincide with the one given above. For functions on open domains, we prove this notion to admit a handier characterization by its Taylor polynomial up to degree ?r?.     

Asymptotically differentiable functions of several variables

We show that if a real function F(x) is defined in an open convex set Q ? Rⁿ and satisfies an asymptotic Lipschitz condition, then it must satisfy a Lipschitz condition. The result is used for investigating asymptotically differentiable functions.     

The structure of module derivations of Banach algebras of differentiable functions

This paper studies the structure and continuity of derivations of the Banach algebra Cⁿ(I) of n times continuously differentiable functions on an interval I into Banach Cⁿ(I)-modules. The structure of derivations into finite dimensional modules is completely determined. The question of when an arbitrary derivation splits into the sum of continuous and singular parts is discussed. An example is constructed of a derivation of C¹(I) which is discontinuous on every dense subalgebra.     

Spectrum of One-Dimensional p-Laplacian with an Indefinite Integrable Weight

Motivated by extremal problems of weighted Dirichlet or Neumann eigenvalues, we will establish two fundamental results on the dependence of weighted eigenvalues of the one-dimensional p-Laplacian on indefinite integrable weights. One is the continuous differentiability of eigenvalues in weights in the Lebesgue spaces L ^γ with the usual norms. Another is the continuity of eigenvalues in weights with respect to the weak topologies in L ^γ spaces. Here 1 ≤ γ ≤ ∞. In doing so, we will give a simpler explanation to the corresponding spectrum problems, with the help of several typical techniques in nonlinear analysis such as the Fréchet derivative and weak* convergence.     

Continuity of iteration and approximation of iterative roots

Motivated by computing iterative roots for general continuous functions, in this paper we prove the continuity of the iteration operators T_n, defined by T_nf=fⁿ. We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If lim_m→∞F_m=F, can we find an iterative root f_m of F_m of order n for each m∈N such that the sequence (f_m) tends to the iterative root of F of order n associated with a given initial function? We not only give the construction of such an approximating sequence (f_m) but also illustrate the approximation of iterative roots with an example. Some remarks are presented in order to compare our approximation with the Hyers-Ulam stability.