CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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$x^{2} \sin (1 / x^{2})$ is differentiable everywhere but not absolutely continuous

Statement refuted

Every everywhere-differentiable function on $[0, 1]$ is absolutely continuous.

Facts & Assumptions

Given: $F (0) = 0$ and $F (x) = x^{2} \sin (1 / x^{2})$ for $0 < x \leq 1$ .

Counterexample

technique · direct

1.1

The difference quotient at $0$ is $x \sin (1 / x^{2})$ , which tends to $0$ ; for $x > 0$ , $F^{'} (x) = 2 x \sin (1 / x^{2}) - 2 \cos (1 / x^{2}) / x$ . Thus $F$ is differentiable everywhere.

givenalgebra

2.1

Put $x_{n} = (π / 2 + n π)^{- 1 / 2}$ . Then $F (x_{n}) = (- 1)^{n} x_{n}^{2}$ , and the finite partitions containing $x_{N} < x_{N - 1} < \dots < x_{0}$ have variation at least $\sum_{n < N} (x_{n}^{2} + x_{n + 1}^{2})$ . This diverges as $N \to \infty$ , so $F$ is not of bounded variation.

step 1.1algebra

3.1

Every absolutely continuous function has bounded variation by $C^{1}$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation. Therefore $F$ is not absolutely continuous.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Donald L. Cohn, Measure Theory, 2nd ed., §6.3, Exercise 6 (standard reference, not scraped)

x2sin⁡(1/x2)x2sin(1/x2) is differentiable everywhere but not absolutely continuous