Analysis

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

See Also:

• Calculus: University first-year level calculus
• Metric Spaces: Sequences and limits in metric spaces
• Multivariable Vector Analysis: Differentiation and integration of multivariable and vector functions

Real Space $\mathbb{R}$

• Three Axioms of Analysis
  • Field Axioms
  • Order Axioms
  • Completeness Axiom
• Archimedean Principle
• Density of Real Numbers
• The Real Set and the Empty Set are Both Open and Closed
• What is an Accumulation Point in the Set of Real Numbers
• Extended Real Number System

Real Sequences

• Why Redefine Limits of Sequences in University Mathematics
• Why Define Convergence of Sequences in a Complex Way in University Mathematics
• Properties of Convergent Real Sequences
• Properties of Divergent Real Sequences
• Monotonic Sequences and the Monotone Convergence Theorem
• Proof of Cantor’s Intersection Theorem
• Proof of the Bolzano–Weierstrass Theorem
• Cauchy Sequences
• Limit Superior and Limit Inferior
• Limit of a Geometric Sequence
• Subsequences
  • Limits of Subsequences and Convergence of Sequences

Series

• Series and Infinite Series
• Definition of Euler’s Number $e$
  • Euler’s Number $e$ is Irrational
• Proof that $\pi$ is Irrational
• Derivation of the Binomial Series
• Maclaurin Expansion of the Fresnel Sine Integral

Continuity

• Limit of a Function: Epsilon-Delta Definition
• The New Definition of Function Continuity in University Mathematics
• Uniform Continuity of Functions

Discontinuity

• Left and Right Limits of Functions
• Classification of Discontinuities
• Piecewise Continuity and Smoothness

Differentiation

• Derivative of a Function Defined on the Real Space
• Differentiability Implies Continuity
• Properties of Differentiable Functions
• Chain Rule of Differentiation
• Definition of Extremum and Its Relation with Derivatives
• Mean Value Theorem
• Continuous but Non-Differentiable Function: Weierstrass Function
• Relation Between Derivative and Monotonicity of Functions
• Leibniz’s Rule for Differentiation

Riemann Integration

Most of the content on integration is based on the PMA textbook, so many proofs are generalized to the Riemann–Stieltjes integral. Setting $\alpha(x) = x$ gives the proofs for the Riemann integral.

• Partition, Riemann Sum, Riemann Integral
• Riemann–Stieltjes Integral
• Refinement of Partitions
• Upper Integral is Greater Than or Equal to Lower Integral
• Necessary and Sufficient Condition for Integrability
  • Continuous Functions are Integrable
  • Monotonic Functions are Integrable
• Non-Integrable Function on a Closed Interval: Dirichlet Function
• Mean 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 an Interval
• Relation Between Functions’ Order and Integrals’ Order
• Integrable Functions and Absolute Values
• Proof of the Mean Value Theorem for Integrals
• Leibniz’s Rule for Integration

Integration and Differentiation

• Fundamental Theorem of Calculus Part 1
• Fundamental Theorem of Calculus Part 2
• Integration by Parts

Curves

• Curves with Measurable Length
• If the Derivative of a Curve is Continuous, the Curve Has Measurable Length

Sequences and Series of Functions

• Pointwise Convergence of Function Sequences
• Uniform Convergence of Function Sequences
  • Uniformly Convergent Sequences are Equivalent to Cauchy Sequences
  • Uniform Convergence and Continuity
  • Uniform Convergence and Differentiability
  • Uniform Convergence and Integrability
• Difference Between Pointwise and Uniform Convergence
• Norm Convergence of Function Sequences
• Algebra of Continuous Function Spaces
• Proof of the Stone–Weierstrass Theorem
• Series of Functions

Power Series

• Power Series
• Radius of Convergence
• Convergence
• Differentiation of Power Series $\dfrac{d}{dx} \left(\sum\limits_{n = 0}^{\infty} c_{n} (x-a)^{n} \right) = \sum\limits_{n = 1}^{\infty} nc_{n} (x-a)^{n-1}$
• Integration of Power Series $\displaystyle \int \sum\limits_{n = 0}^{\infty} c_{n} (x-a)^{n} dx = \sum\limits_{n = 0}^{\infty} \dfrac{c_{n}}{n+1} (x-a)^{n+1} + C$
• Cauchy Product: Product of Two Convergent 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)