Author's Preface
Preview

Chapter 1 Real Number System
1.1 Is √2 an irrational number?
1.2 Mathematical Fundamentals
Practice problems
1.3 Completeness Axiom
Practice problems
1.4 Corollary of the completeness axiom
Practice problems
1.5 riders
Practice problems
1.6 Cantor's theorem
1.7 In conclusion

Chapter 2 Sequences and Series
2.1 Can an infinite series be rearranged?
2.2 Limits of sequences
Practice problems
2.3 Summary of Limits
Practice problems
2.4 A Taste of Monotonic Convergence and Infinite Series
Practice problems
2.5 Subsequences and the Bolzano-Weierstrass theorem
Practice problems
2.6 Cauchy convergence test
Practice problems
2.7 Properties of infinite series
Practice problems
2.8 Multiplication of double and infinite series
2.9 In conclusion

Chapter 3 Topological Properties of Real Numbers
3.1 Cantor set
3.2 Open and closed sets
Practice problems
3.3 Compact sets
Practice problems
3.4 Complete sets and connected sets
Practice problems
3.5 Bertrand's theorem
3.6 In conclusion

Chapter 4 Limits and Continuity of Functions
4.1 Dirichlet function and Tomé function
4.2 Limits of functions
Practice problems
4.3 Continuous functions
Practice problems
4.4 Continuous functions defined on compact sets
Practice problems
4.5 Intermediate value arrangement
Practice problems
4.6 Set of discontinuities
4.7 In conclusion

Chapter 5 Derivatives
5.1 Is the derivative continuous?
5.2 Properties of derivatives and intermediate values
Practice problems
5.3 Mean value summary
Practice problems
5.4 Functions that are continuous but not differentiable everywhere
5.5 In conclusion

Chapter 6 Function Sequences and Function Series
6.1 Exponentiation of power series
6.2 Uniform convergence of a series of functions
Practice problems
6.3 Uniform convergence and differentiation
Practice problems
6.4 Function series
Practice problems
6.5 Power series
Practice problems
6.6 Taylor series
Practice problems
6.7 Weierstrass approximation theorem
6.8 In conclusion

Chapter 7 Riemann Integral
7.1 How is integral defined?
7.2 Riemann integral
Practice problems
7.3 Integral of functions with discontinuities
Practice problems
7.4 Properties of integration
Practice problems
7.5 Fundamental Theorem of Differential and Integral Calculus
Practice problems
7.6 Lebesgue test for Riemann integrability
7.7 In conclusion

Chapter 8 One Step Further
8.1 Generalized Riemann Integral
8.2 Distance Space and Berkeley Category Theorem
8.3 Euler sum
8.4 Creating a Factorial Function
8.5 Fourier series
8.6 Creating real numbers using rational numbers

References
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Analysis is a discipline that developed by laying the mathematical foundation for differential and integral calculus, and has influenced all fields of mathematics.
Moreover, analysis is also used in fields that deal deeply with functions, such as physics, engineering, and economics, so it can be said that analysis has meaning in various fields.
However, the first hurdle that greets beginners in analysis who have just begun to study 'real' mathematics is not so easy.
It is common to be at a loss when redefining the concept of limits, which we took for granted in differential and integral calculus.
If you are still having trouble with existing hermeneutics textbooks, start with “First Steps in Hermeneutics.”


〈First Steps in Hermeneutics〉 (original title: Understanding Analysis, 2nd edition) is a one-semester textbook on single-variable analysis, and both the first and second editions have received great reviews from international readers.
The first edition clearly presents the reasons why a beginner in hermeneutics should study hermeneutics, to the point where it is introduced as "a dangerous book because it explains things so well."
It is also a recommended analysis textbook for undergraduate students by the Mathematical Association of America (MAA).

Through this book, you will learn that hermeneutics is not simply a sophisticated refinement of differential and integral calculus.
Let's look at why the proofs we learned in calculus were not rigorous.
You can sense how complex mistakes are, how subtle the differences in the various convergence conditions are, and what intellectual joy lies behind the infinite paradoxes.

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You can have a mystical experience of completing hermeneutics by going back and forth between intuition and argument.
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