TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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Every derivative is Henstock–Kurzweil integrable and satisfies Newton–Leibniz

Statement

Let $a < b$ , let $F : [a, b] \to R$ be differentiable in the domain-relative sense, including one-sided endpoint derivatives, and put $f = F^{'}$ . Every derivative is Henstock–Kurzweil integrable and its integral equals the endpoint increment:

$\int_{a}^{b} f = F (b) - F (a) .$

No boundedness or prior integrability of $f$ is assumed.

Every derivative is Henstock–Kurzweil integrable and its integral is the endpoint increment. Every derivative is Henstock–Kurzweil integrable and evaluates by endpoint difference.

Every derivative is Henstock–Kurzweil integrable and its integral equals the endpoint increment. Every derivative is Henstock–Kurzweil integrable.

Facts & Assumptions

Given: The differentiable function $F$ and $f = F^{'}$ .

[L1]

The derivative at a point is the limit of the difference quotient there, including the domain-relative endpoint cases (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

[L2]

For every positive real $r$ , there is a natural $n \geq 1$ with $1 / n < r$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L3]

Every nonempty subset of $N$ has a least element (The well-ordering principle).

[L4]

Finite sums telescope: $\sum_{i = 1}^{m} (c_{i} - c_{i - 1}) = c_{m} - c_{0}$ (Laws of finite sums and finite products).

Proof

technique · direct

1.1

Given $ε > 0$ , put $η = ε / (2 (b - a))$ . For each $ξ \in [a, b]$ , let $N (ξ)$ be the least natural $n \geq 1$ such that the derivative estimate with error $η$ holds whenever $0 < ∣ y - ξ ∣ < 1 / n$ in the domain. Differentiability and [L2] make this set nonempty, and [L3] makes its least element unique; hence $δ (ξ) = 1 / N (ξ)$ is a gauge defined without an uncountable choice.

givenL1L2L3

2.1

For a fine tagged cell, split $F (v_{i}) - F (u_{i})$ at its tag, apply the two estimates from step 1.1, sum over all cells, and telescope by [L4]; the total error is below $2 η (b - a) = ε$ , proving the displayed HK value.

step 1.1L4algebra∎

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Sources

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Sections 1.2 and 1.21 (standard reference, not scraped)