ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Differentiation on C^1[0,1] is closed and unbounded in the supremum norm

Example

With supremum norms, $D : C^{1} [0, 1] \subset C [0, 1] \to C [0, 1]$ , $D f = f^{'}$ , has closed graph but is unbounded.

Facts & Assumptions

Given: $f_{n} \to f$ and $f_{n}^{'} \to g$ uniformly, with $f_{n} \in C^{1} [0, 1]$ .

Verification

technique · direct

1.1given

Newton--Leibniz Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative gives $f_{n} (t) - f_{n} (0) = \int_{0}^{t} f_{n}^{'} (s) d s$ . Passing to uniform limits yields $f (t) - f (0) = \int_{0}^{t} g (s) d s$ .

2.1step 1.1

Since $g$ is continuous, the integral function is differentiable with derivative $g$ ; hence $f \in C^{1}$ and $D f = g$ . The graph (The graph of a linear operator with a linear domain) is closed.

3.1given∎

For integers $n \geq 1$ , put $f_{n} (t) = \sin (n t) / n$ . Then $∥ f_{n} ∥_{\infty} \leq 1 / n$ whereas $∥ D f_{n} ∥_{\infty} = 1$ , so $D$ is unbounded.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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

Teschl, Topics in Real and Functional Analysis, Problem 4.8 (standard reference, not scraped)