CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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 $x\ne0$ , and $f(0,y)=0$ .

[L1]

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

[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 $(ta,tb)$ , the restriction is $t^2b(a^2+b^2)/a$ when $a\ne0$ 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 $t\ne0$ , $f(t^3,t)=1+t^4\to1$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 19 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

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

The map y(x2+y2)/xy(x^2+y^2)/xy(x2+y2)/x off the line x=0x=0x=0, extended by zero on that line, has every directional derivative zero at the origin but is discontinuous there