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

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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: $f(x,y)=xy(x^2-y^2)/(x^2+y^2)$ 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 $y\ne0$ , $f(0,y)=0$ , so $f_y(0,0)=0$ ; for $x\ne0$ , $f_y(x,0)=x$ , hence $f_{xy}(0,0)=1$ .

givenalgebra

1.2

For $x\ne0$ , $f(x,0)=0$ , so $f_x(0,0)=0$ ; for $y\ne0$ , $f_x(0,y)=-y$ , hence $f_{yx}(0,0)=-1$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 43 results over 15 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)