Alphabeta Math

The Fundamental Theorems of Calculus

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✓ 9 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Remark

Roadmap through the three strengths of FTC I and the five Riemann strengths of FTC II

Corollary

Assuming Countable Choice, the integral function of a Riemann-integrable function is Lipschitz and differentiable almost everywhere, with derivative equal to the integrand there

Theorem

Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative

Corollary

Newton–Leibniz remains valid across finitely many exceptional interior points when the primitive is continuous

Theorem

Botsko's theorem: if F is continuous on [a,b], F'(x)=f(x) off a countable subset of (a,b), and f is Riemann integrable, then ∫ₐᵇ f=F(b)-F(a)

Theorem

Integration by parts for continuous factors with Riemann-integrable extensions of their interior derivatives

Theorem

Substitution for a continuous inner map with a Riemann-integrable extension of its interior derivative, without monotonicity or injectivity

Theorem

For an integrable f, the one-sided derivatives of F(x)=∫ₐˣ f equal the corresponding one-sided limits of f; at a jump they are unequal

Theorem

Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral

Theorem

The two FTC forms for a Riemann–Stieltjes integral with a C¹ integrator

Sources