CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$x^{2} y / (x^{4} + y^{2})$ tends to zero on every line through the origin but not along $y = x^{2}$

Statement refuted

If a function tends to its proposed value along every straight line through a point, then it is continuous at that point.

Facts & Assumptions

Given: $f (0, 0) = 0$ and $f (x, y) = x^{2} y / (x^{4} + y^{2})$ away from the origin.

[L1]

A vector-valued map is continuous at $a$ when its limit at $a$ equals its value there (Vector-valued functions $f : A \to R^{m}$ , their limits and continuity, with the dictionary to the metric notions).

Counterexample

technique · direct

1.1

On a line $(x, y) = (t a, t b)$ with $b \neq 0$ , $f (t a, t b) = t a^{2} b / (t^{2} a^{4} + b^{2}) \to 0$ ; for $b = 0$ the restriction is identically zero.

2.1

Along the parabola $(x, y) = (t, t^{2})$ with $t \neq 0$ , $f (t, t^{2}) = 1 / 2$ .

step 1.1algebra

3.1

The nonlinear path tends to the origin but its values do not tend to $0$ , so [L1] shows that $f$ is not continuous there despite all straight-line tests.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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

J. Knisley, Multivariable Calculus, §2.2 (standard reference, not scraped)

x2y/(x4+y2)x2y/(x4+y2) tends to zero on every line through the origin but not along y=x2y=x2