Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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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x2y/(x4+y2)x^2y/(x^4+y^2) tends to zero on every line through the origin but not along y=x2y=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)=0f(0,0)=0 and f(x,y)=x2y/(x4+y2)f(x,y)=x^2y/(x^4+y^2) away from the origin.

[L1]

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

Counterexample

technique · direct
1.1

On a line (x,y)=(ta,tb)(x,y)=(ta,tb) with b≠0b\ne0, f(ta,tb)=ta2b/(t2a4+b2)→0f(ta,tb)=t a^2b/(t^2a^4+b^2)\to0; for b=0b=0 the restriction is identically zero.

L1
2.1

Along the parabola (x,y)=(t,t2)(x,y)=(t,t^2) with t≠0t\ne0, f(t,t2)=1/2f(t,t^2)=1/2.

step 1.1algebra
3.1

The nonlinear path tends to the origin but its values do not tend to 00, so [L1] shows that ff 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 131 results over 22 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