CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-29

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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.
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If $f, g$ are integrable on $[a, b]$ then so are $∣ f ∣$ , $f^{2}$ , $f g$ , $\max (f, g)$ and $\min (f, g)$ , and $∣ \int_{a}^{b} f ∣ \leq \int_{a}^{b} ∣ f ∣$

Statement

Let $a < b$ be reals and let $f, g : [a, b] \to R$ be integrable (The lower and upper Darboux integrals of a bounded $f$ on $[a, b]$ as $\sup_{P} L (f, P)$ and $\inf_{P} U (f, P)$ , Darboux integrability as their equality, and the notation $\int_{a}^{b} f$ ). Then:

$∣ f ∣$ , $f^{2}$ and $f g$ are integrable on $[a, b]$ (Absolute value in an ordered field, Integer powers $a^{m}$ );
$\max (f, g)$ and $\min (f, g)$ , defined pointwise (Maximum and minimum of a set), are integrable on $[a, b]$ ;
the triangle inequality for the integral: $∣ \int_{a}^{b} f ∣ \leq \int_{a}^{b} ∣ f ∣ .$

Claim 3 is stated with $a < b$ and is not orientation-invariant. For $u > v$ the right-hand side is $- \int_{v}^{u} ∣ f ∣ \leq 0$ while the left-hand side is $\geq 0$ , so the inequality as written is false there. The form valid for every pair $u, v$ on which $f$ is integrable (The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ) is

$∣ \int_{u}^{v} f ∣ \leq ∣ \int_{u}^{v} ∣ f ∣ ∣,$

and that is the form the estimates below on this page use whenever the limits are not known to be in increasing order.

The converse of claim 1 fails. Integrability of $∣ f ∣$ does not give integrability of $f$ ; the witness is on the companion page.

Facts & Assumptions

Given: Reals $a < b$ and integrable $f, g : [a, b] \to R$ .

[L1]

If $u$ is integrable on $[a, b]$ with values in $[m, M]$ and $φ$ is continuous on $[m, M]$ , then $φ \circ u$ is integrable (If $f$ is integrable on $[a, b]$ with values in $[m, M]$ and $φ$ is continuous on $[m, M]$ , then $φ \circ f$ is integrable); an integrable function is bounded, so such $m$ and $M$ exist (For bounded $f$ on $[a, b]$ and a partition $P$ : the infimum $m_{i}$ and supremum $M_{i}$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L (f, P) = \sum_{i} m_{i} Δ_{i}$ and $U (f, P) = \sum_{i} M_{i} Δ_{i}$ , Lower bound, bounded below, bounded set, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L2]

Sums and scalar multiples of integrable functions are integrable, with $\int_{a}^{b} (λ u + ν w) = λ \int_{a}^{b} u + ν \int_{a}^{b} w$ (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ).

[L3]

If $u \leq w$ pointwise on $[a, b]$ and both are integrable then $\int_{a}^{b} u \leq \int_{a}^{b} w$ (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[L4]

The absolute value $t \mapsto ∣ t ∣$ , the square $t \mapsto t^{2}$ and every polynomial function are continuous on every subset of $R$ (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, claims 2 and 5, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point).

[L5]

For reals $s, t$ : $\max {s, t} = (s + t + ∣ s - t ∣) \cdot 2^{- 1}$ and $\min {s, t} = (s + t - ∣ s - t ∣) \cdot 2^{- 1}$ , and $s t = ((s + t)^{2} - (s - t)^{2}) \cdot 4^{- 1}$ (Maximum and minimum of a set, Absolute value in an ordered field, Ordered field, Integer powers $a^{m}$ ).

[L6]

Absolute value: $- ∣ t ∣ \leq t \leq ∣ t ∣$ , and $∣ t ∣ \leq c$ follows from $- c \leq t \leq c$ (Basic properties of the absolute value, Absolute value in an ordered field).

[L7]

With oriented limits, $\int_{v}^{u} u = - \int_{u}^{v} u$ and $\int_{u}^{u} u = 0$ (The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ).

[L8]

Ordered-field arithmetic: adding constants and multiplying by positive reals preserve inequalities, and the order is total and transitive (Ordered field, Complete ordered field (least-upper-bound property)).

Proof

technique · direct

1.1

$f$ is bounded, so fix reals $m \leq M$ with $f [[a, b]] \subseteq [m, M]$ ; the same for $g$ , and for $f + g$ and $f - g$ , which are integrable by [L2].

givenL1L2choose

1.2

The maps $t \mapsto ∣ t ∣$ and $t \mapsto t^{2}$ are continuous on any closed bounded interval, by [L4].

2.1

By [L1] applied with $φ (t) = ∣ t ∣$ to $f$ , to $g$ and to $f - g$ , the functions $∣ f ∣$ , $∣ g ∣$ and $∣ f - g ∣$ are integrable.

step 1.1step 1.2L1

2.2

By [L1] applied with $φ (t) = t^{2}$ to $f$ , to $f + g$ and to $f - g$ , the functions $f^{2}$ , $(f + g)^{2}$ and $(f - g)^{2}$ are integrable.

step 1.1step 1.2L1

3.1

By [L5], $f g = ((f + g)^{2} - (f - g)^{2}) \cdot 4^{- 1}$ pointwise, so $f g$ is integrable by [L2]; this completes claim 1.

step 2.1step 2.2L2L5

3.2

By [L5], $\max (f, g) = (f + g + ∣ f - g ∣) \cdot 2^{- 1}$ and $\min (f, g) = (f + g - ∣ f - g ∣) \cdot 2^{- 1}$ pointwise, so both are integrable by [L2]; this is claim 2.

step 2.1L2L5

3.3

Claim 3. By [L6], $- ∣ f ∣ \leq f \leq ∣ f ∣$ pointwise on $[a, b]$ , and all three functions are integrable by step 2.1 and [L2].

step 2.1L2L6

4.1

By [L3] applied twice, $- \int_{a}^{b} ∣ f ∣ \leq \int_{a}^{b} f \leq \int_{a}^{b} ∣ f ∣$ , using $\int_{a}^{b} (- ∣ f ∣) = - \int_{a}^{b} ∣ f ∣$ from [L2].

step 3.3L2L3

5.1

Hence $∣ \int_{a}^{b} f ∣ \leq \int_{a}^{b} ∣ f ∣$ by [L6], which is claim 3.

step 4.1L6

6.1

The oriented form. For $u = v$ both sides are $0$ by [L7]; for $u < v$ it is claim 3 on $[u, v]$ ; and for $u > v$ both $\int_{u}^{v} f$ and $\int_{u}^{v} ∣ f ∣$ are the negatives of the corresponding integrals over $[v, u]$ by [L7], so the two absolute values are unchanged and claim 3 on $[v, u]$ gives the inequality.

step 5.1L7L8∎

Remarks

Every integrability clause comes from one theorem plus linearity. The only input that produces integrability is If $f$ is integrable on $[a, b]$ with values in $[m, M]$ and $φ$ is continuous on $[m, M]$ , then $φ \circ f$ is integrable, with Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ recombining the pieces; claim 3 additionally uses If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , which is the one place an inequality between integrals is needed. The identities of [L5] are algebra, and they are what turns a statement about composing with $∣ \cdot ∣$ and $(\cdot)^{2}$ into statements about products and lattice operations. In particular no new estimate on Darboux sums is made here.
The polarisation identity is used, and it is why $f^{2}$ comes first. There is no direct route from integrability of $f$ and of $g$ to integrability of $f g$ through the composition theorem, because $(s, t) \mapsto s t$ is a function of two variables and the theorem composes with one. Writing $f g$ through squares of sums and differences reduces it to the one-variable case.
The inequality of claim 3 is the integral analogue of the triangle inequality, and like it, it can be strict: for $f (x) = x$ on $[- 1, 1]$ the left-hand side is $0$ and the right-hand side is $1$ .
Forward reference, orientation only. The witness refuting the converse of claim 1 is A function that is not Riemann integrable although $∣ f ∣$ is ↗ on the companion page; nothing above depends on it.

Depends on

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Sources

Riemann integral (Wikipedia) (standard reference, not scraped)
W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 6 (standard reference, not scraped)
Carnegie Mellon 21-269, Riemann integration notes (standard reference, not scraped)
Encyclopedia of Mathematics, Integral calculus (standard reference, not scraped)

If f,gf,g are integrable on [a,b][a,b] then so are ∣f∣∣f∣, f2f2, fgfg, max⁡(f,g)max(f,g) and min⁡(f,g)min(f,g), and ∣∫abf∣≤∫ab∣f∣​∫ab​f​≤∫ab​∣f∣