CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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.

Every real-analytic function is infinitely differentiable

Statement

Every real-analytic function on an open subset $U\subseteq\mathbb R$ is infinitely differentiable at every point of $U$ .

Facts & Assumptions

Given: A real-analytic function $f:U\to\mathbb R$ .

[L1]

Near every $c\in U$ , $f$ equals a convergent power series centred at $c$ (A real-analytic function on an open subset of $\mathbb{R}$ is locally represented by a convergent real power series).

[L2]

A power-series sum is infinitely differentiable inside its radius (A power-series sum is infinitely differentiable inside its radius and satisfies $a_n=f^{(n)}(c)/\iota(n!)$ at its centre).

Proof

technique · direct

1.1

Fix $c\in U$ and choose the local representing power series supplied by [L1].

givenL1choose

2.1

By [L2], that series, hence $f$ , has derivatives of every order on a neighbourhood of $c$ . Since $c$ was arbitrary, $f$ is smooth on $U$ .

step 1.1L2∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 55 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Analytic function, Encyclopedia of Mathematics (standard reference, not scraped)
E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)