University of Isfahan Mathematics and Society 2345-6493 1 3 2000 09 01 The derivative à la Carathéodory The derivative à la Carathéodory 25 29 7862 10.22108/msci.2016.7862 FA Mohammad Sal Moslehian Ferdowsi University of Mashhad Journal Article 1970 01 01 This paper is a translation of the the following paper into Persian: [Kuhn, Stephen, The derivative à la Carathéodory, Amer. Math. Monthly, 98 no. 1 (1991) 40--44]. Abstract Translator: According to Carathéodory, a function $f$ is differentiable at a point $a \in D_f$ if there exists a function $\varphi$ continuous at $a$ such that for each $x$ in an open interval $U$ containing $a$ it holds that $f(x)-f(a)=\varphi(x)(x-a)$. In this paper we investigate this definition, prove that it is equivalent to the usual notion of differentiability and show that it can be used to prove some elementary theorems in the topic of derivative in calculus. This paper is a translation of the the following paper into Persian: [Kuhn, Stephen, The derivative à la Carathéodory, Amer. Math. Monthly, 98 no. 1 (1991) 40--44]. Abstract Translator: According to Carathéodory, a function $f$ is differentiable at a point $a \in D_f$ if there exists a function $\varphi$ continuous at $a$ such that for each $x$ in an open interval $U$ containing $a$ it holds that $f(x)-f(a)=\varphi(x)(x-a)$. In this paper we investigate this definition, prove that it is equivalent to the usual notion of differentiability and show that it can be used to prove some elementary theorems in the topic of derivative in calculus.

https://math-sci.ui.ac.ir/article_7862_317b8f8c392fe54bb10aff4ce32a44eb.pdf