多元函数的可微性

给出了多元函数可微性的等价条件：利用这个条件可以简便有效地判别多元函数的可微性。     

关于多元函数可微性的一个注记

研究了多元函数的可微性,给出了多元函数连续、可微的一些条件.     

一类二元函数连续与可微条件的归纳与推广

多元函数的可导性、连续性、可微性是多元函数微分学的重要内容.本文基于多元微分学教学实践,归纳了一类二元函数的可偏导、连续、可微的充分条件,并对此类二元函数做一般性推广,得到了推广函数连续与可微的充分条件.     

一个关于多元函数可微的定理

大家知道,多元函数的可微与可导是两个概念,但如果一个多元函数的所有偏导数在某一点都存在并连续,则它一定在该点可微.本文给出了一个在较弱条件下关于多元函数可微的定理.     

微分模式

１．论述ｄｘ与△ｘ的关系；ｄｘ是从△ｘ→０转化而来．比△ｘ更为抽象；２对多元函数引出几．种可微性概念，更加细致地刻划其可微性。     

F值函数的微分及其在FNN中的应用

本文引进了集值函数的ｓ－可微和模糊值（Ｆ值）函数的Ｆｓ－可微概念。给出了这两种可微性的几个判别条件。最后研究并得到了一类模糊神经网络（ＦＮＮ）的Ｆｓ－可微性和连续性。     

关于多元函数可微性的一个注记

对全微分的存在性进行了讨论．给出了某类多元函数z=f（P）在点Po处可微的一个判别方法     

函数可微性的几何刻画与反函数可微性的几何证明

类似于二元函数可微性的几何刻画,本文从几何的角度给出了一元函数可微的充分必要条件.利用这一等价性条件得到了反函数可微的一个更为简洁直观的几何证明.     

多元复合函数求导链式法则证明的注记

"当自变量增量趋向于0时,中间变量增量可能取到0"是多元复合函数求导链式法则证明过程中的主要困难.本文通过补充定义的方法克服该难点,在外层函数可微、内层函数可导这一最弱条件下严格化了证明.     

模糊值函数的凸性与次可微性

在Goetschel-Voxman所定义的序关系下,首先讨论了模糊值函数的凸性,得到了凸模糊值函数的若干充分条件,并证明了凸模糊值函数的Jensen不等式;其次,讨论了凸模糊值函数的次可微性,给出了次微分的若干重要性质,并得到了次可微条件下取得最优解的充分必要条件以及若干个次可微的充分条件.     

Differentiability in several variables revisited

This paper introduces a concept of differentiability for functions in several variables which has a very natural definition and shares all the properties one expects of a notion of differentiability, but is shown to be weaker than the usual concept of differentiability defined as local linearizability. The exact connection between the two different concepts is exhibited.     

La différentiabilité dans le calcul fractionnaire

In this work we give a general concept of differentiability of order α ε [0, 1] for functions of one variable, and then for functions of several variables, in Nishimoto fractional calculus.     

Differentiability of Peano derivatives

Peano differentiability is a notion of higher-order Fréchet differentiability. H. W. Oliver gave sufficient conditions for the Peano derivative to be a Fréchet derivative in the case of functions of a real variable. Here we generalize this theorem to functions of several variables.

     

On p-adic differentiability

A p-adic-valued function on the p-adic integers has a continuous derivative, Mahler showed, whenever its interpolation coefficients decay at a certain rate. It is shown here that Mahler's decay condition is equivalent to the strict differentiability of the function. There is a discussion of the Banach-space structure of the space of strictly differentiable functions. It is shown, moreover, that there is no rate of decay common to all functions with continuous derivative. Specifically, given any decay condition, there exists a function with derivative identically zero, whose interpolation coefficients decay more slowly.     

二元函数的可微性研究

研究二元函数的可微性,得到两个充分条件,它们与已有的判别二元函数可微性的充分条件相比更为简单,因而使得应用更为简便.     

Convex fuzzy mapping with differentiability and its application in fuzzy optimization

The concepts of differentiability, convexity, generalized convexity and minimization of a fuzzy mapping are known in the literature. The purpose of this present paper is to extend and generalize these concepts to fuzzy mappings of several variables using Buckley–Feuring approach for fuzzy differentiation and derive Karush–Kuhn–Tucker condition for the constrained fuzzy minimization problem.     

The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality. Communicated by Steven Krantz     

Characterizations of <Emphasis Type="Italic">r</Emphasis>-Convex Functions

This paper discusses some properties of r-convexity and its relations with some other types of convexity. A characterization of convex functions in terms of r-convexity is given without assuming differentiability. The concept of strict r-convexity is introduced. For a twice continuously differentiable function f, it is shown that the strict r-convexity of f is equivalent to a certain condition on ∇ ² f. Further, it is shown that this condition is satisfied by quasiconvex functions satisfying a less stringent condition.     

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)     

On differentiability of ruin functions under Markov-modulated models

This paper analyzes the continuity and differentiability of several classes of ruin functions under Markov-modulated insurance risk models with a barrier and threshold dividend strategy, respectively. Many ruin related functions in the literature, such as the expectation and the Laplace transform of the Gerber–Shiu discounted penalty function at ruin, of the total discounted dividends until ruin, and of the time-integrated discounted penalty and/or reward function of the risk process, etc, are special cases of the functions considered in this paper. Continuity and differentiability of these functions in the corresponding dual models are also studied.