CounterexampleConstruction: Literature-sourcedVerification: Not suppliedaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A $C^{1}$ map whose critical values have positive measure

Statement refuted

False claim: every $C^{1}$ map has critical values of measure zero.

Whitney's classical construction gives a $C^{1}$ map $f : R^{2} \to R$ whose critical values contain a set of positive measure. That source-backed phenomenon is the concrete counterexample behind Sard's theorem does not hold for every $C^{1}$ map; this item records the example honestly without pretending to reconstruct Whitney's construction inside the current page.

Depends on

Sard's theorem does not hold for every $C^1$ map

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Encyclopedia of Mathematics, Sard theorem (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)

A C1C1 map whose critical values have positive measure