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
- 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
Series
- Series and Infinite Series
- Definition of Euler’s Number $e$
- 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
- 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
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
- 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
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)
All posts
- Limit of Subsequence and Convergence of Sequence
- Subsequence
- Uniform Convergence of a Sequence of Functions and Differentiability
- Uniform Convergence and Integrability of Function Series
- Radius of Convergence of a Power Series
- Differentiation of Power Series
- Convergence of Power Series
- Integration of Power Series
- Three Axioms of Analysis: 1 Field Axioms
- Three Axioms of Analysis: The Second Order Axiom
- Principle of Archimedes in Analysis
- Three Axioms of Analysis: The Axiom of Completeness
- Proof of the Density of Real Numbers
- Mean Value Theorem for Integrals
- Fresnel Sine Integral's Maclaurin Series Expansion
- Partition, Riemann Sum, Riemann Integral
- Riemann-Stieltjes Integral
- Segmentation
- Upper integral is greater than or equal to lower integral.
- Necessary and Sufficient Conditions for Riemann(-Stieltjes) Integrability
- Continuous Functions are Riemann-Stieltjes Integrable
- Monotone Functions are Riemann-Stieltjes Integrable
- Integrability is Preserved in the Composition with Continuous Functions
- Integrability is Preserved in the Multiplication of Two Functions
- Proof of Leibniz's Theorem
- Series, Infinite Series
- Local Lipschitz Condition
- Continuity in Every Piece, Smoothness in Every Segment
- Mean of Function Values
- Convergence of Norms of Function Sequences
- Power Series
- Cauchy Product: The Product of Two Convergent Power Series
- Binomial Series Derivation
- Algebra of the Space of Continuous Functions
- Proof of the Stone-Weierstrass Theorem
- Proof that Pi is an Irrational Number
- The Euler Constant e is an Irrational Number
- Functions That Cannot Be Integrated over a Closed Interval: The Dirichlet Function
- Pointwise Convergence of Function Sequences
- Uniform Convergence of Function Series
- The Accumulation Point in the Set of Real Numbers
- The set of real numbers and the empty set are both open and closed.
- The Difference between Pointwise Convergence and Uniform Convergence of Functions
- Functions of Series
- Continuous but Not Differentiable Functions: Weierstrass Function
- Redefining the Limits of Sequences in University Mathematics
- The Reason for Intricately Defining the Convergence of Sequences in University Mathematics
- Cantor's Intersection Theorem
- Bolzano-Weierstrass Theorem
- Cauchy Sequence
- Limits Supremum and Limits Infimum
- Epsilon-Delta Argument
- Newly Defined Continuous Functions in University Mathematics
- Uniform Continuity of Functions
- Differentiation of Functions Defined in Real Number Space
- Extended Real Number System
- Leibniz Integral Rule
- Linearity of Riemann(-Stieltjes) Iintegral
- Integrable Functions and Absolute Values
- Riemann-Stieltjes Integrability is Preserved within an Interval
- The Fundamental Theorem of Calculus in Analysis
- The Relationship Between the Size of Integrals Based on the Order of Functions
- Properties of Converging Real Sequences
- If Differentiable, Then Continuous
- Differentiable Function Properties
- Definition and Relationship of Extremum in Analysis and Differential Coefficients
- The Chain Rule of Differentiation in Analysis
- Mean Value Theorem in Analysis
- The Relationship between Derivatives and the Increasing/Decreasing of Functions
- Limits from the Left and the Right Strictly Defined in Analysis
- Classification of Discontinuities
- The Fundamental Theorem of Calculus in Analysis
- Integration by Parts
- Measuring Curves: A Guide to Length
- If the Derivative of a Curve is Continuous, the Curve Can Be Measured
- Necessary and Sufficient Condition for Uniform Convergence
- The Definition of Euler's Constant, the Natural Number e
- Monotone Sequence and Monotone Convergence Theorem
- Properties of Divergent Real Sequences
- Uniform Convergence and Continuity of Function Sequences
- Definition of Improper Integrals