TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The two FTC forms for a Riemann–Stieltjes integral with a $C^{1}$ integrator

Statement

Let $a < b$ . Suppose $α : [a, b] \to R$ is continuous, differentiable on $(a, b)$ , and its derivative extends to a continuous function $q : [a, b] \to R$ .

If $f : [a, b] \to R$ is Riemann integrable and continuous at $c$ , then $A (x) : = \int_{a}^{x} f d α$ is differentiable at $c$ and $A^{'} (c) = f (c) q (c)$ .
If $f$ is Riemann integrable, $G : [a, b] \to R$ is continuous and differentiable on $(a, b)$ , and $G^{'} (x) = f (x) q (x)$ there, then $\int_{a}^{b} f d α = G (b) - G (a)$ .

Endpoint derivatives in clause 1 are relative. Clause 2 does not divide by $q$ and remains valid where $q$ vanishes.

Facts & Assumptions

Given: The functions in the statement.

[L1]

For a continuous integrator whose interior derivative extends continuously as $q$ , every Riemann-integrable $f$ is Riemann--Stieltjes integrable and $\int_{u}^{v} f d α = \int_{u}^{v} f q$ on each closed subinterval (A continuously differentiable integrator reduces Stieltjes integration to ordinary integration).

[L2]

A continuous function on a compact interval is Riemann integrable, and products of Riemann-integrable functions are integrable; hence $f q$ is integrable because $q$ is continuous (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion, If $f, g$ are integrable on $[a, b]$ then so are $∣ f ∣$ , $f^{2}$ , $f g$ , $\max (f, g)$ and $\min (f, g)$ , and $∣ \int_{a}^{b} f ∣ \leq \int_{a}^{b} ∣ f ∣$ ).

[L3]

The integral function of an integrable function is differentiable at each continuity point, with derivative equal to the integrand (The first fundamental theorem: if $f$ is integrable on $[a, b]$ and continuous at $c$ , then $F^{'} (c) = f (c)$ ; in particular a continuous $f$ has $F$ as a primitive).

[L4]

A continuous function with an interior derivative admitting an integrable extension satisfies Newton--Leibniz (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative).

Proof

technique · reduction

1.1

By [L1], $A (x) = \int_{a}^{x} f (t) q (t) d t$ . The product $f q$ is integrable by [L2] and is continuous at $c$ .

givenL1L2

1.2

Under the hypotheses of clause 2, [L1] and [L2] give $\int_{a}^{b} f d α = \int_{a}^{b} f q$ , while $f q$ is an integrable extension of the interior derivative of $G$ .

givenL1L2

2.1

Applying [L3] to the ordinary integral function in step 1.1 gives $A^{'} (c) = f (c) q (c)$ , including either relative endpoint case.

step 1.1L3

3.1

Applying [L4] to $G$ gives $\int_{a}^{b} f q = G (b) - G (a)$ , and step 1.2 proves the second clause without any division by $q$ .

step 1.2L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

47 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

T. M. Apostol, Mathematical Analysis, 2nd ed., Chapter 7 (standard reference, not scraped)

The two FTC forms for a Riemann–Stieltjes integral with a C1C1 integrator