CounterexampleConstruction: Literature-sourcedVerification: 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.
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Peano's function has unequal mixed partials at the origin

Statement refuted

Refuted: existence of both mixed partial derivatives at a point forces them to be equal.

Facts & Assumptions

Given: $f (x, y) = x y (x^{2} - y^{2}) / (x^{2} + y^{2})$ away from $(0, 0)$ and $f (0, 0) = 0$ .

[L1]

Clairaut--Schwarz requires continuous second partial derivatives on a neighbourhood, not merely their existence at one point (Clairaut--Schwarz theorem for continuous second partial derivatives).

Counterexample

Proof

technique · direct

1.1

For $y \neq 0$ , $f (0, y) = 0$ , so $f_{y} (0, 0) = 0$ ; for $x \neq 0$ , $f_{y} (x, 0) = x$ , hence $f_{x y} (0, 0) = 1$ .

givenalgebra

1.2

For $x \neq 0$ , $f (x, 0) = 0$ , so $f_{x} (0, 0) = 0$ ; for $y \neq 0$ , $f_{x} (0, y) = - y$ , hence $f_{y x} (0, 0) = - 1$ .

givenalgebra

2.1

Thus the two mixed partials exist and differ, while the continuity hypothesis in [L1] fails.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)