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Mathematical Analysis Fundamentals by Agamirza Bashirov

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Chapter 7

Differentiation

Abstract

Chapter 7 considers a differentiation issue for real-valued functions of real variable. It starts with the definition of derivative and its discussion. A special attention is paid to the relation between the continuity and differentiation. A continuous nowhere differentiable function due to Van der Waerden is discussed. Rules of differentiation are proved. Mean value theorems in the standard, Cauchy and Taylor’s forms and implications from them are discussed. Intermediate value theorem for derivatives is proved and it is emphasized that the derivative function may be discontinuous. Differential equations are introduced and Peano’s theorem on existence of solution for differential equations is proved. Banach spaces ...

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