LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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.

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A rectangular second difference equals a mixed partial times the side lengths

Statement

Let $R$ be a closed axis-parallel rectangle and suppose that $f_x$ and $f_{xy}$ exist on an open neighbourhood of $R$ . If $(x_0,y_0),(x_1,y_1)$ are opposite corners of a nondegenerate subrectangle of $R$ , then some $\xi$ strictly between $x_0,x_1$ and some $\eta$ strictly between $y_0,y_1$ satisfy

$f(x_1,y_1)-f(x_1,y_0)-f(x_0,y_1)+f(x_0,y_0)=(x_1-x_0)(y_1-y_0)f_{xy}(\xi,\eta).$

Facts & Assumptions

Given: The stated open-neighbourhood hypotheses and a nondegenerate subrectangle of $R$ .

[L1]

After ordering its two endpoints, the one-variable mean-value theorem gives $g(v)-g(u)=(v-u)g'(c)$ for a function continuous on the closed interval and differentiable on its interior, with $c$ strictly between the endpoints (The mean value theorem, as the case $g(x) = x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a < b$ and differentiable on $(a,b)$ there is $c \in (a,b)$ with $f(b) - f(a) = f'(c)(b-a)$ ).

Proof

technique · direct

1.1

Apply [L1] in the $x$ variable to $x\mapsto f(x,y_1)-f(x,y_0)$ . The stated existence of $f_x$ on an open neighbourhood gives the required one-variable regularity, and the rectangle difference is $(x_1-x_0)(f_x(\xi,y_1)-f_x(\xi,y_0))$ .

L1givenchoose

2.1

Apply [L1] in the $y$ variable to $y\mapsto f_x(\xi,y)$ . Since $f_{xy}$ exists on an open neighbourhood, this one-variable map is continuous on the closed interval and differentiable on its interior. This yields $(y_1-y_0)f_{xy}(\xi,\eta)$ and proves the formula.

step 1.1L1givenchoose∎

Depends on

Used by

Peano's mixed-partial theorem from continuity of one mixed partial Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Mixed partial derivatives (Eremenko) (standard reference, not scraped)