Alphabeta MathThe Fundamental Theorems of Calculus: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Fundamental Theorems of Calculus
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
G(x)=x² sin(1/x) has a bounded derivative discontinuous at 0 that is nevertheless Riemann integrable, and Newton–Leibniz evaluates its integral
Counterexample
Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable
Example
A bounded increasing integrand discontinuous at every rational has an integral function nondifferentiable at every rational in (0,1)
Example
Thomae's integrand is discontinuous at every rational, yet its integral function is identically zero and differentiable everywhere
Example
The indicator of {1,1/2,1/4,1/8,…} is discontinuous at 0, but its integral function has derivative 0=f(0) there
Counterexample