Titlebook: Real Analysis and Applications; Fabio Silva Botelho Textbook 2018 Springer Nature Switzerland AG 2018 real analysis.Implicit function theo

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ng it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis..978-3-030-08750-0978-3-319-78631-5

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易于交谈

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Metric SpacesIn this chapter we present the main definitions and results related to metric spaces. Standard topics such as open, closed, and compact sets are developed in detail. In the final section we prove the Heine–Borel theorem.

沙草纸

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DerivativesThis chapter presents the main definitions and results related to derivatives for one variable real functions. Standard topics such as the derivative proprieties, the mean value theorem, and Taylor expansion are developed in detail. The inverse function theorem and related derivative for such a one real variable case is also addressed.

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Real Numbershe subsequent sections the definitions of integer and rational numbers are introduced. In the final sections the existence of the real field is proven through the definition of cut. We finish such a chapter with a study about countable and uncountable sets.

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Real Sequences and Seriesences and their applicability in the proof of root and ratio tests for convergence of series. The comparison criterion for series is also extensively addressed and applied to a great variety of situations.