False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-21

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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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FALSE: every differentiable function has a continuous derivative

Statement

False claim: if $F : R \to R$ is differentiable, then its derivative $F^{'}$ is continuous on $R$ .

Facts & Assumptions

Given: The universal claim in the Statement.

[L1]

The function $F (0) = 0$ and $F (x) = x^{2} \sin (1 / x^{2})$ for $x \neq 0$ is differentiable on $R$ , with $F^{'} (0) = 0$ , and $F^{'}$ is unbounded on every neighbourhood of zero ( $x^{2} \sin (1 / x^{2})$ has an unbounded, non-Riemann-integrable derivative).

Refutation

technique · contradiction

1.1assume-contra

Suppose, for contradiction, that every differentiable real function has a continuous derivative.

1.2L1algebra

The function in [L1] is differentiable on all of $R$ , but its derivative is unbounded on every neighbourhood of zero and therefore cannot be continuous at zero.

2.1step 1.1step 1.2L1discharge-contradiction∎

Step 1.1 makes the derivative in step 1.2 continuous, a contradiction. Therefore the claim is false.

Depends on

$x^2\sin(1/x^2)$ has an unbounded, non-Riemann-integrable derivative

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

John K. Hunter, An Introduction to Real Analysis, Example 8.10 (standard reference, not scraped)
Adam Coffman, Yifei Pan, and Yuan Zhang, Continuous Solutions of Nonlinear Cauchy–Riemann Equations and Pseudoholomorphic Curves in Normal Coordinates, Remark 3.6 (standard reference, not scraped)