What this book should do
- Introduce concepts of analysis in its generalized form
- Although definitions are defined in generalized form, offer its real analysis correspondence
- Investigate the consequences of real analysis definitions 
# Real analysis
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1. real numbers
- Field of real numbers
- Total order of real numbers
- Preservation of order under addition
- Preservation of order under multiplication
- Order is complete (Least upper bound property)
- Archimedian property
- R is a 1D Euclidean space
- linear-algebra: triangle inequality
- bounded set
- upper bound
- lower bound
- supremum
- infimum
- absolue value function (metric)
1. Sequences
- sequence
- monotone sequence
- subsequence
- convergent sequence
- Cauchy sequence
- Bolzano-Weierstrass theorem (BWT)
- compact space
- complete space
- squeeze theorem
- limit supremum
- limit infimum
2. Series
- series
- absolutely convergent series
- conditionally convergent series
- harmonic series
- Euler-Mascheroni constant
- geometric series
- partial sum test
- comparison test
- limit comparison test
- ratio rest
- Leibniz test
- nth root test
- P test
- Cauchy condensation test
- integral test
3. Continuous functions
- limit
- one-sided limit
- continuous function
- uniformly continuous function
- lipschitz continuous function
- extreme value theorem
- intermediate value theorem
4. Differentiation
- differentiable function
- derivative
- product rule
- chain rule
- quotient rule
- maxima
- Rolle's theorem
- mean value theorem (MVT)
- generalized mean value theorem (GMVT)
- L"Hopital's rule
- inverse function theorem
- convex function
- concave funtion
- concave funtion
- global maximum
- global minimum
- local maximum
- local minimum 
- power series
- Taylor series
- Taylor's theorem
- analytic function
- C^k spaces
- smooth function
- continuously differentiable function
- first derivative test
- second derivative test

5. Riemann integration
- interval partitoon
- uniform partition 
- mesh
- Riemann integrable function
- Riemann's criterion
- fundamental theorem of calculus (FTC)
- integration by parts
- integration by substitution
- mean value theorem of integrals
- improper Riemann integral
- Cauchy principle value (CPV)
- arc length of a function
5. Sequences of functions
- sequence of functions
- pointwise convergent sequence of functions
- uniformly convergent sequence of functions
- uniformly Cauchy sequence of functions
- Weierstrass M test
- Arzela's bounded convergence theorem
- Weierstrass approximation theorem
- Stone-Weierstrass theorem
- characteristic function
- Dirichlet function
- Dini's theorem
6. Advanced integration
series method
Leibniz integral rule
7. \part{Special real functions}
As consequences of the many definitions introduced in real analysis (and therefore differential equations), many special functions that cannot be expressed as finite combinations of the 4 arithmetic operations arise.

Many of these functions exhibit interesting properties that either arise from their definition or can be proven using the methods of real analysis.


- \chapter{Sine function}
\[\sin(0)=0, \sin''=-\sin\]
\[ \sin(x)= \sum^{\infty}_{n=0} (-1)^{n}\frac{x^{2n+1}}{(2n+1)!} \]
\chapter{Cosine function}
\[\cos(0)=1, \cos''=-\cos\]
\[ \cos(x)= \sum^{\infty}_{n=0} (-1)^{n}\frac{x^{2n}}{(2n)!} \]
\section{}
\chapter{Exponential function}
\[ \exp(0)=1, \exp'=\exp \]
\[ \exp(x)= \sum^{\infty}_{n=0} \frac{x^n}{n!} \]
\chapter{Natural logarithm}
\[ \ln (e^x)=x\]
\section{Series on $[1,2]$}

- \chapter{Gamma function}
- \section{Euler's reflection formula}
- \chapter{Beta function}
- \chapter{Bessel functions}
- \chapter{Modified Bessel functions}
- \chapter{Airy functions}
- \chapter{Weierstrass function}
- \chapter{Riemann zeta function}

The following property due to Euler connects this function intimately to number theory.
- \section{Euler's product formula}
- \chapter{Riemann xi function}
- \chapter{Dirichlet eta function}
8. Lebesgue integration

7. Asymptotic analysis
- Bachmann-Landau notations
- small O
- big O
- big theta
- asymptotic equivalence
- big omega
- small omega

Appendix: basel problem