TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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

Statement

Let $f$ have $f_{x y}$ in a neighbourhood of $(a, b)$ , with $f_{x y}$ continuous at $(a, b)$ , and let $f_{y x} (a, b)$ exist. Then $f_{x y} (a, b) = f_{y x} (a, b)$ .

Facts & Assumptions

Given: The hypotheses in the statement.

[L1]

When $f_{x}$ and $f_{x y}$ exist on a neighbourhood of a rectangle, its rectangular second difference is the product of the side lengths and a value of $f_{x y}$ (A rectangular second difference equals a mixed partial times the side lengths).

Proof

technique · direct

1.1

For sufficiently small nonzero $h, k$ , apply [L1] to the rectangle with corners $(a, b)$ and $(a + h, b + k)$ . After division by $h k$ , continuity of $f_{x y}$ at $(a, b)$ makes the limit, as $h, k \to 0$ , equal to $f_{x y} (a, b)$ .

L1givenalgebra

2.1

For fixed nonzero $h$ , first let $k \to 0$ in the same rectangle quotient; it becomes $(f_{y} (a + h, b) - f_{y} (a, b)) / h$ . Letting $h \to 0$ gives the defining quotient for $f_{y x} (a, b)$ .

step 1.1givenalgebra

3.1

The two limits are equal, proving $f_{x y} (a, b) = f_{y x} (a, b)$ .

step 1.1step 2.1∎

Depends on

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

Used by

Clairaut--Schwarz theorem for continuous second partial derivatives Theorem

Dependency tree · two levels

3 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

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