False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: every continuous real function is differentiable somewhere

Statement

False claim: every continuous function $f : R \to R$ is differentiable at at least one real point.

Facts & Assumptions

Given: The universal claim in the Statement.

[L1]

There is a sequence of smooth functions converging uniformly on $R$ to a continuous function which is differentiable at no real point (A uniform limit of smooth functions need not be differentiable anywhere).

Refutation

technique · contradiction

1.1

Suppose, for contradiction, that every continuous real function is differentiable somewhere.

assume-contra

1.2

Let $W$ be the continuous nowhere-differentiable function whose existence is asserted by [L1].

L1choose

2.1

The assumption in step 1.1 makes $W$ differentiable at some real point, contradicting [L1]. Therefore the universal claim is false.

step 1.1step 1.2L1discharge-contradiction∎

Depends on

A uniform limit of smooth functions need not be differentiable anywhere

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Jeff Calder, Weierstrass's Non-Differentiable Function, historical introduction and Theorem 1 (standard reference, not scraped)