ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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 four Dini derivatives of x sin(1/x) at 0 take two distinct values

Example

Let $f (0) = 0$ and $f (x) = x \sin (1 / x)$ for $x \neq 0$ . Then

$D^{+} f (0) = D^{-} f (0) = 1, D_{+} f (0) = D_{-} f (0) = - 1.$

In particular the four Dini derivatives at $0$ are not all equal, so $f$ is not differentiable at $0$ .

Facts & Assumptions

Given: The function $f (x) = x \sin (1 / x)$ for $x \neq 0$ and $f (0) = 0$ .

[A1]

The symbols are those of the statement.

Verification

technique · direct

1.1givenalgebra

For $h \neq 0$ , $\frac{f (h) - f (0)}{h} = \sin (1 / h)$ . Choose $h_{n} = \frac{1}{π / 2 + 2 π n}$ and $k_{n} = \frac{1}{3 π / 2 + 2 π n}$ , both positive. Then $\sin (1 / h_{n}) = 1$ and $\sin (1 / k_{n}) = - 1$ , so the upper and lower right Dini derivatives are at least $1$ and at most $- 1$ respectively. Since $∣ \sin (1 / h) ∣ \leq 1$ always, we get $D^{+} f (0) = 1$ and $D_{+} f (0) = - 1$ .

2.1step 1.1

Choose instead $h_{n}^{-} = - \frac{1}{3 π / 2 + 2 π n}$ and $k_{n}^{-} = - \frac{1}{π / 2 + 2 π n}$ . Then $\sin (1 / h_{n}^{-}) = 1$ and $\sin (1 / k_{n}^{-}) = - 1$ , so the upper and lower left Dini derivatives are $D^{-} f (0) = 1$ and $D_{-} f (0) = - 1$ .

3.1step 1.1step 2.1∎

Therefore the four Dini derivatives are exactly the four stated values and are not all equal.

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