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首页> 外文期刊>Journal of Mathematical Sciences >NONSMOOTH DIFFERENTIATION AS AN OPERATOR
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NONSMOOTH DIFFERENTIATION AS AN OPERATOR

机译：算符的非光滑差分

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The classical Newton-Leibnitz formula (in its "school variant") can be written down as an equivalence of equalitiesf(x) = ∫_a~xF(t)dt + f(a) (1.1)and f'(x) = F(x) (1.2) if we suppose that F is continuous and f is continuously differentiate. These assumptions, which are far from those where (1.1) and (1.2) have a sense, induced researchers to reconsider integral and differential operations. For example, it was one of the stimuli for Lebesgue, who had attained the whole completeness of the equivalence (1.1)-(1.2) with the Lebesgue integral in (1.1) and relevant extension (functions integrated by Lebesgue in the space L_1) of classical differentiation over closure: any integral function F and any differential function f can appear in (1.1)-(1.2). The same completeness can be attained without extending the notion of the integral, that is, remaining in the structure of the Riemann integral and extending just the notion of the derivative. If there is a function F that is Riemann-integrable, where (1.1) can postulate (1.2), it is enough to claim that the function f is differentiable. We examine this in detail. First, we will speak about the functions of the one-dimensional variable. It is widely known that a bounded function F : [a, b] → R is Riemann-integrable if and only if it is continuous almost everywhere. It is clear that the object of the examinations, connected with (1.1), must be not individual functions but classes of equivalence, where F is equivalent to G if F and G coincide almost everywhere. We denote a set of equivalent classes (which are continuous functions bounded almost everywhere) by C_(ae). Apparently, it is a vector space above R, a ring, a lattice.
机译：牛顿-莱伯尼茨经典公式（在其“学派形式”中）可以记为等式f（x）=∫_a〜xF（t）dt + f（a）（1.1）和f'（x）=如果我们假设F是连续的并且f是连续微分的，则F（x）（1.2）。这些假设与（1.1）和（1.2）有意义的假设相距甚远，这促使研究人员重新考虑积分和微分运算。例如，这是Lebesgue的刺激之一，他的（1.1）-（1.2）的Lebesgue积分和（1.1）的相关扩展（Lebesgue的功能在L_1空间中的积分）已经完全等效。关于闭包的经典微分：（1.1）-（1.2）中可以出现任何积分函数F和任何微分函数f。无需扩展积分的概念，即保留在黎曼积分的结构中，而仅扩展导数的概念，就可以实现相同的完整性。如果存在一个黎曼可积分的函数F，其中（1.1）可以假定（1.2），则足以断言该函数f是可微的。我们将对此进行详细研究。首先，我们将讨论一维变量的功能。众所周知，当且仅当有界函数F：[a，b]→R几乎在任何地方都是连续的，它是黎曼可积的。显然，与（1.1）有关的检查对象必须不是单个功能，而是等价类，如果F和G几乎在任何地方都重合，则F等于G。我们表示由C_（ae）组成的一组等效类（几乎是无处不在的连续函数）的集合。显然，它是R上方的向量空间，环，晶格。

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