On Dini and Approximate Dini Derivates of Typical Continuous Functions
نویسنده
چکیده
In the thirties, Banach, Mazurkiewicz and Jarnnk found relations connecting Dini derivates of a typical continuous function on 0; 1] at all points of (0; 1). We prove, answering a question of K. M. Garg, that there are no further relations of this sort. An analogous result is proved also for approximate Dini derivates. The aim of this note is to present relatively simple proofs of these results. An article containing an improvement of these results in several directions (with a considerably more complicated proof) is in preparation.
منابع مشابه
Titchmarsh theorem for Jacobi Dini-Lipshitz functions
Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lip...
متن کاملFundamental Theorem of Calculus for Lebesgue Integration
The existing proofs of the Fundamental theorem of calculus for Lebesgue integration typically rely either on the Vitali–Carathéodory theorem on approximation of Lebesgue integrable functions by semi-continuous functions (as in [3, 9, 12]), or on the theorem characterizing increasing functions in terms of the four Dini derivates (as in [6, 10]). Alternatively, the theorem is derived using the Pe...
متن کاملSubdifferentials of Performance Functions and Calculus of Coderivatives of Set-valued Mappings
The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gâteaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in arbitrary Banach spaces. The key elem...
متن کاملDini Derivative and a Characterization for Lipschitz and Convex Functions on Riemannian Manifolds
Dini derivative on Riemannian manifold setting is studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.
متن کاملStability of Solutions to Impulsive Caputo Fractional Differential Equations
Stability of the solutions to a nonlinear impulsive Caputo fractional differential equation is studied using Lyapunov like functions. The derivative of piecewise continuous Lyapunov functions among the nonlinear impulsive Caputo differential equation of fractional order is defined. This definition is a natural generalization of the Caputo fractional Dini derivative of a function. Several suffic...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999