Derivative of a function, Generalization of the Caratheodory Definition

Subject Areas : Statistics

Ali Parsian ¹

1 - Department of Mathematics, Tafresh Universiy, , Iran.Abstract

Received: 2020-10-30 Accepted : 2022-03-07 Published : 2022-03-21

Keywords: قضیة تیلور, قضیه مقدار میانگین کوشی, قضیة رول, قضیة مقدار میانگین,

Abstract :

ABSTRACT:The current definition of the derivative makes the set of differentiable functions much smaller than the set of continuous functions, such that most of the real single variable functions are not differentiable and the surveying the rate of their growth is not possible with the available definition. In the present paper, using Caratheodory definition, we extend the set of differentiable functions by introducing a definition for generalized differentiation of a single variable function and its generalized derivative, in such a way that the validity of the basic theorems of this theory such as Rolle's theorem, Cauchy's mean value theorem, mean value theorem and Taylor's theorem would be hold. Finally we give some examples.----------------------------ABSTRACT:The current definition of the derivative makes the set of differentiable functions much smaller than the set of continuous functions, such that most of the real single variable functions are not differentiable and the surveying the rate of their growth is not possible with the available definition. In the present paper, using Caratheodory definition, we extend the set of differentiable functions by introducing a definition for generalized differentiation of a single variable function and its generalized derivative, in such a way that the validity of the basic theorems of this theory such as Rolle's theorem, Cauchy's mean value theorem, mean value theorem and Taylor's theorem would be hold. Finally we give some examples.

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