Analysis

This category deals with real sequences ${x_n}: \mathbb{N} \to \mathbb{R}$ and real functions $f: \mathbb{R} \to \mathbb{R}$.

For content lacking in continuity and differentiation, see the 'Metric Space' category.

Multivariable functions and vector functions are covered in the 'Vector Analysis' category.

Real Space $\mathbb{R}$

• Three axioms of analysis
  • Field axioms
  • Order axioms
  • Completeness axioms
• Archimedes’ principle
• Density of the real numbers
• The real number set and the empty set are both open and closed
• Accumulation points in the real number set
• Extended real number system

Sequences of Real Numbers

• Why the limit of a sequence is redefined in university mathematics
• Why the convergence of sequences is defined more complexly in university mathematics
• Properties of converging real sequences
• Properties of diverging real sequences
• Monotonic sequences and monotone sequence theorem
• Proof of Cantor’s intersection theorem
• Proof of Bolzano-Weierstrass theorem
• Cauchy sequences
• Limit superior, limit inferior
• Limits of geometric sequence
• Subsequence
  • Limits of Subsequences and Convergence of Sequences

Series

• Series, Infinite series
• Definition of Euler’s constant $e$
  • Euler’s constant $e$ is irrational
• Proof that pi is irrational
• Derivation of the binomial series
• Maclaurin expansion of Fresnel sine integrals

Continuity

• Limit of a function: Epsilon-delta argument
• New definition of continuity in university mathematics
• Uniform continuity of functions

Discontinuity

• Left-hand and right-hand limits of functions
• Classification of discontinuities
• Piecewise continuity, piecewise smoothness

Differentiation

• Differentiation of functions defined in real space
• If differentiable, then continuous
• Properties of differentiable functions
• Chain rule of differentiation
• Definition of a maximum and its relationship with the derivative
• Mean value theorem
• Continuous but non-differentiable functions: Weierstrass function
• Relationship between the derivative and the increase/decrease of a function
• Leibniz’s rule for differentiation

Riemann Integration

Content on integration primarily references PMA textbooks, so many proofs generalize to Riemann-Stieltjes integration. Setting $\alpha (x) = x$ yields proofs equivalent to those for Riemann integration.

• Partition, Riemann sum, Riemann integration
• Riemann-Stieltjes integration
• Refinement
• Upper integral is greater than or equal to lower integral
• Necessary and sufficient conditions for integrability
  • Continuous functions are integrable
  • Monotonic functions are integrable
• Functions not integrable on a closed interval: Dirichlet function
• Average value of a function
• Definition of improper integrals

Properties of Integration

• Integration is linear
• Integrability is preserved under composition with continuous functions
• Integrability is preserved under multiplication of two functions
• Integrability is preserved within intervals
• Integral comparison based on function magnitude
• Integrable functions and their absolute values
• Proof of the mean value theorem for integrals
• Leibniz rule for integration

Integration and Differentiation

• Fundamental theorem of calculus 1
• Fundamental theorem of calculus 2
• Integration by parts

Curves

• Curves that can be measured
• If the derivative of a curve is continuous, the curve can be measured

Sequences and Series of Functions

• Pointwise convergence of function sequences
• Uniform convergence of function sequences
  • A Sequence that Uniformly Converges and a Cauchy Sequence are Equivalent
  • Uniform Convergence and Continuity
  • Uniform Convergence and Differentiation
  • Uniform Convergence and Integration
• Difference between pointwise and uniform convergence
• Norm convergence of function sequences
• Algebra of continuous function spaces
• Proof of the Stone-Weierstrass theorem
• Series of function sequences

Power Series

• Power series
• Radius of convergence
• Convergence
• Derivation of power series
• Cauchy product: Product of two converging power series

Miscellaneous

• Local Lipschitz condition

References

• James Stewart, Daniel Clegg, and Saleem Watson, Calculus (early transcendentals, 9E)
• William R. Wade, An Introduction to Analysis (4th Edition, 2010)
• Walter Rudin, Principles of Mathmatical Analysis (3rd Edition, 1976)