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

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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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The map $y (x^{2} + y^{2}) / x$ off the line $x = 0$ , extended by zero on that line, has every directional derivative zero at the origin but is discontinuous there

Statement refuted

If every directional derivative at a point is zero, then the function is continuous there.

Facts & Assumptions

Given: $f (x, y) = y (x^{2} + y^{2}) / x$ for $x \neq 0$ , and $f (0, y) = 0$ .

[L1]

The directional derivative is the derivative of $t \mapsto f (a + t v)$ at zero (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[L2]

Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity (Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity).

Counterexample

technique · direct

1.1

Along $(t a, t b)$ , the restriction is $t^{2} b (a^{2} + b^{2}) / a$ when $a \neq 0$ and is $0$ when $a = 0$ ; [L1] therefore gives zero directional derivative in every direction.

L1L2

2.1

Along the curve $(x, y) = (t^{3}, t)$ with $t \neq 0$ , $f (t^{3}, t) = 1 + t^{4} \to 1$ .

step 1.1L2algebra

3.1

Thus $f$ is discontinuous at the origin, and by [L2] it cannot be totally differentiable there either.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)

The map y(x2+y2)/xy(x2+y2)/x off the line x=0x=0, extended by zero on that line, has every directional derivative zero at the origin but is discontinuous there